Logiweb codex of sup2

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sup2, LeqTotality(R), PositiveToLeft(Eq), ExpZero(Exact), SameExp(Base), SameExp(Indu), SameExp, (1/2)(x + y) − x = (1/2)(y − x), y − (1/2)(x + y) = (1/2)(y − x), BSzero(Exact), SameBS(2)(Base), SameBS(2)(Indu), SameBS(2), NegativeToLeft(Less)(1term), BS(+1), BSbound(Exact)(Base), BSbound(Exact)(Indu), BSbound(Exact), BSbound, UStelescope(Zero)(Exact), SameTelescope(2)(Base), SameTelescope(2)(Indu), SameTelescope(2), Exp(+1), PositiveBase(Base), PositiveBase(Indu), PositiveBase, TelescopeNumerical(Base), TelescopeNumerical(Indu), TelescopeNumerical, (+1)IsPositive(N), DistributionOut(Minus), PositiveToRight(Eq)(1term), SameSeries(NumDiff), ToNegatedDoubleImply, AddNegatedAll, (A)to( E )(Imply), (E)to( A )(Imply), (E )to( A)(Imply), ToNegatedAEA, Three2threeTerms(R), LessNeq(F)(Helper), LessNeq(F), LessNeq(R), x = x + (y − y)(R), x = x + y − y(R), SubtractEquations(R), NeqAddition(R), PositiveToRight(Less)(R), PositiveToRight(Less)(1term)(R), LeqNeqLess(R), SubLeqLeft(R), ToLeq(Advanced)(R), LeqLessTransitivity(R), NegativeToLeft(Eq)(R), NegativeToRight(Less)(R), !! == Symmetry, SwitchTerms(x <= y − z), Plus0Left(R), PositiveToRight(Eq)(R), EqAdditionLeft(R), Three2twoTerms(R), To!! ==, PositiveToRight(Less)(1term), (A )to( E), NegativeToRight(Eq)(R), NegativeToRight(Eq)(1term)(R), DoubleMinus(R), UniqueNegative(R), SubtractEquationsLeft(R), EqNegated(R), NeqNegated(R), −0 = 0(R), NegativeNegated(R), FromLeqGeq(R), 0 <= |x|(R), PositiveNegated(R), AddEquations(R), Times(−1)(R), Times(−1)Left(R), −x − y = −(x + y)(R), LessTotality(R), SameNumerical(R), MinusNegated(R), PositiveNumerical(R), SignNumerical(+)(R), NonnegativeNumerical(R), NegativeNumerical(R), LeqNegated(R), LessNegated(R), SubLeqRight(R), FromLess(R), DistributionOut(R), x ∗ 0 + x = x(R), x ∗ 0 = 0(R)(fff), SignNumerical(R), NumericalDifference(R), x <= |x|(R), USlimitIsUpperBound(Helper), USlimitIsUpperBound, (−1) ∗ (−1) + (−1) ∗ 1 = 0(R), (−1) ∗ (−1) = 1(R), 0 < 1Helper(R), 0 < 1(R), ExpZero(Exact)(R), PositiveBase(R)(Base), Three2twoFactors(R), x = x ∗ y ∗ (1/y)(R), NeqMultiplication(R), LessTransitivity(R), 0 < 2(R), SameExp(R)(Base), SameExp(R)(Indu), SameExp(R), SubNeqLeft(R), SubNeqRight(R), NonzeroFactors(R), NonnegativeFactors(R), PositiveFactors(R), LessDivision(R), 0 < 1/2(R), PositiveToRight(Eq)(1term)(R), Exp(+1)(R), PositiveBase(R)(Indu), PositiveBase(R), −x ∗ y = −(x ∗ y)(R), PositiveToLeft(Eq)(R), Times1Left(R), x + x = 2 ∗ x(R), (1/2)x + (1/2)x = x(R), DistributionOut(Minus)(R), (1/2)(x + y) − x = (1/2)(y − x)(R), IntervalSize(R)(Base), LessMultiplicationLeft(R), NegativeToLeft(Less)(R), NegativeToLeft(Less)(1term)(R), y − (1/2)(x + y) = (1/2)(y − x)(R), IntervalSize(R)(Indu), IntervalSize(R), XSlessUS(R), USdecreasing(+1)(R), ExpUnbounded(Base), ExpUnbounded(Indu), ExpUnbounded, 1 <= x + 1(N), ExpNonzero(Base), ExpNonzero(Indu), ExpNonzero, ExpNonzero(2), HalfBase(Base), HalfBase(Indu), MultiplyEquations(R), NonreciprocalToRight(Eq)(1term)(R), PositiveNonzero(R), NonzeroProduct(2)(R), HalfBase, Three2threeFactors(R), x ∗ y = zBackwards(R), PositiveInverted(R), ReciprocalToRight(Less)(R), ReciprocalToRight(Less)(1term)(R), NonreciprocalToLeft(Less)(R), 1 < x ∗ y(R), SwitchFactors(1/x < y)(R), SmallHalving, IntervalSize(anyPositive), USdecreasing(+n)(Base), USdecreasing(+n)(Indu), USdecreasing(+n), USdecreasing, LeqAdditionLeft(R), ToNotLess(R), LimitOfUSIsLeq, SubtractEquations(Less)(R), SubtractEquationsLeft(Less)(R), LessNegated(Negative)(R), FromNegatedAnd(Imply), RemoveDoubleNeg(Consequent), FromNotUpperBound, LeqNUB, USlimitIsLeastUpperBound(Helper), USlimitIsLeastUpperBound, ExistMP3, GreaterPositive(N), ysFClose(Helper), ysFClose, ysFCauchy(Helper), ysFCauchy, from <<==, to <<==, NonnegativeNumerical(F), NegativeNumerical(F), tester1, tester2, tester3, tester4, tester5, tester6,

sup2

[sup2 →^{prio}

Preassociative [sup2], [base], [bracket ∗ end bracket], [big bracket ∗ end bracket], [ $ ∗ $ ], [flush left [∗]], [x], [y], [z], [[∗ ▷◁ ∗]], [[∗ →^{∗} ∗]], [pyk], [tex], [name], [prio], [∗], [T], [if(∗, ∗, ∗)], [[∗ ⇒^{∗} ∗]], [val], [claim], [⊥], [f(∗)], [(∗)^{I}], [F], [\underline{0}], [\underline{1}], [\underline{2}], [\underline{3}], [\underline{4}], [\underline{5}], [\underline{6}], [\underline{7}], [\underline{8}], [\underline{9}], [0], [1], [2], [3], [4], [5], [6], [7], [8], [9], [a], [b], [c], [d], [e], [f], [g], [h], [i], [j], [k], [l], [m], [n], [o], [p], [q], [r], [s], [t], [u], [v], [w], [(∗)^{M}], [If(∗, ∗, ∗)], [array{∗} ∗ end array], [l], [c], [r], [empty], [⟨∗ | ∗:= ∗⟩], [M(∗)], [Ũ(∗)], [U(∗)], [U^{M}(∗)], [apply(∗, ∗)], [apply_{1}(∗, ∗)], [identifier(∗)], [identifier_{1}(∗, ∗)], [arrayplus(∗, ∗)], [array-remove(∗, ∗, ∗)], [array-put(∗, ∗, ∗, ∗)], [array-add(∗, ∗, ∗, ∗, ∗)], [bit(∗, ∗)], [bit_{1}(∗, ∗)], [rack], ["vector"], ["bibliography"], ["dictionary"], ["body"], ["codex"], ["expansion"], ["code"], ["cache"], ["diagnose"], ["pyk"], ["tex"], ["texname"], ["value"], ["message"], ["macro"], ["definition"], ["unpack"], ["claim"], ["priority"], ["lambda"], ["apply"], ["true"], ["if"], ["quote"], ["proclaim"], ["define"], ["introduce"], ["hide"], ["pre"], ["post"], [E(∗, ∗, ∗)], [E_{2}(∗, ∗, ∗, ∗, ∗)], [E_{3}(∗, ∗, ∗, ∗)], [E_{4}(∗, ∗, ∗, ∗)], [lookup(∗, ∗, ∗)], [abstract(∗, ∗, ∗, ∗)], [⌈∗⌉], [M(∗, ∗, ∗)], [M_{2}(∗, ∗, ∗, ∗)], [M^{∗}(∗, ∗, ∗)], [macro], [s_{0}], [zip(∗, ∗)], [assoc_{1}(∗, ∗, ∗)], [(∗)^{p}], [self], [[∗ ¨= ∗]], [[∗ ˙= ∗]], [[∗ ´= ∗]], [[∗ =^{pyk} ∗]], [[∗ =^{tex} ∗]], [[∗ =^{name} ∗]], [Priority table[∗]], [M̃_{1}], [M̃_{2}(∗)], [M̃_{3}(∗)], [M̃_{4}(∗, ∗, ∗, ∗)], [M̃(∗, ∗, ∗)], [Q̃(∗, ∗, ∗)], [Q̃_{2}(∗, ∗, ∗)], [Q̃_{3}(∗, ∗, ∗, ∗)], [Q̃^{∗}(∗, ∗, ∗)], [(∗)], [(∗)], [display(∗)], [statement(∗)], [[∗]^{·}], [[∗]^{−}], [aspect(∗, ∗)], [aspect(∗, ∗, ∗)], [⟨∗⟩], [tuple_{1}(∗)], [tuple_{2}(∗)], [let_{2}(∗, ∗)], [let_{1}(∗, ∗)], [[∗ =^{claim} ∗]], [checker], [check(∗, ∗)], [check_{2}(∗, ∗, ∗)], [check_{3}(∗, ∗, ∗)], [check^{∗}(∗, ∗)], [check_{2}^{∗}(∗, ∗, ∗)], [[∗]^{·}], [[∗]^{−}], [[∗]^{∘}], [msg], [[∗ =^{msg} ∗]], [<stmt>], [stmt], [[∗ =^{stmt} ∗]], [HeadNil′], [HeadPair′], [Transitivity′], [⊥ ⊥], [Contra′], [T′_{E}], [L_{1}], [\underline{∗}], [A], [B], [C], [D], [E], [F], [G], [H], [I], [J], [K], [L], [M], [N], [O], [P], [Q], [R], [S], [T ], [U], [V], [W], [X], [Y], [Z], [⟨∗ | ∗:= ∗⟩], [⟨^{∗}∗ | ∗:= ∗⟩], [∅], [Remainder], [(∗)^{v}], [intro(∗, ∗, ∗, ∗)], [intro(∗, ∗, ∗)], [error(∗, ∗)], [error_{2}(∗, ∗)], [proof(∗, ∗, ∗)], [proof_{2}(∗, ∗)], [S(∗, ∗)], [S^{I}(∗, ∗)], [S^{⊳} (∗, ∗)], [S_{1}^{⊳} (∗, ∗, ∗)], [S^{E}(∗, ∗)], [S_{1}^{E}(∗, ∗, ∗)], [S^{+}(∗, ∗)], [S_{1}^{+}(∗, ∗, ∗)], [S^{−}(∗, ∗)], [S_{1}^{−}(∗, ∗, ∗)], [S^{∗}(∗, ∗)], [S_{1}^{∗}(∗, ∗, ∗)], [S_{2}^{∗}(∗, ∗, ∗, ∗)], [S^{@}(∗, ∗)], [S_{1}^{@}(∗, ∗, ∗)], [S^{⊢}(∗, ∗)], [S_{1}^{⊢}(∗, ∗, ∗, ∗)], [S⊢ ⊢(∗, ∗)],

[S⊢ ⊢ ; 1(∗, ∗, ∗, ∗)], [S^{i.e.}(∗, ∗)], [S_{1}^{i.e.}(∗, ∗, ∗, ∗)], [S_{2}^{i.e.}(∗, ∗, ∗, ∗, ∗)], [S^{∀}(∗, ∗)],

[S_{1}^{∀}(∗, ∗, ∗, ∗)], [S^{;}(∗, ∗)], [S_{1}^{;}(∗, ∗, ∗)], [S_{2}^{;}(∗, ∗, ∗, ∗)], [T (∗)], [claims(∗, ∗, ∗)], [claims_{2}(∗, ∗, ∗)], [<proof>], [proof], [[Lemma ∗: ∗]], [[Proof of ∗: ∗]], [[∗ lemma ∗: ∗]], [[∗ antilemma ∗: ∗]], [[∗ rule ∗: ∗]], [[∗ antirule ∗: ∗]], [verifier], [V_{1}(∗)], [V_{2}(∗, ∗)], [V_{3}(∗, ∗, ∗, ∗)], [V_{4}(∗, ∗)], [V_{5}(∗, ∗, ∗, ∗)], [V_{6}(∗, ∗, ∗, ∗)], [V_{7}(∗, ∗, ∗, ∗)], [Cut(∗, ∗)], [Head_{⊕}(∗)], [Tail_{⊕}(∗)], [rule_{1}(∗, ∗)], [rule(∗, ∗)], [Rule tactic], [Plus(∗, ∗)], [[Theory ∗]], [theory_{2}(∗, ∗)], [theory_{3}(∗, ∗)], [theory_{4}(∗, ∗, ∗)], [HeadNil′′], [HeadPair′′], [Transitivity′′], [Contra′′], [HeadNil], [HeadPair], [Transitivity], [Contra], [T_{E}], [ragged right], [ragged right expansion ], [parm(∗, ∗, ∗)], [parm^{∗}(∗, ∗, ∗)], [inst(∗, ∗)], [inst^{∗}(∗, ∗)], [occur(∗, ∗, ∗)], [occur^{∗}(∗, ∗, ∗)], [unify(∗ = ∗, ∗)], [unify^{∗}(∗ = ∗, ∗)], [unify_{2}(∗ = ∗, ∗)], [L_{a}], [L_{b}], [L_{c}], [L_{d}], [L_{e}], [L_{f}], [L_{g}], [L_{h}], [L_{i}], [L_{j}], [L_{k}], [L_{l}], [L_{m}], [L_{n}], [L_{o}], [L_{p}], [L_{q}], [L_{r}], [L_{s}], [L_{t}], [L_{u}], [L_{v}], [L_{w}], [L_{x}], [L_{y}], [L_{z}], [L_{A}], [L_{B}], [L_{C}], [L_{D}], [L_{E}], [L_{F}], [L_{G}], [L_{H}], [L_{I}], [L_{J}], [L_{K}], [L_{L}], [L_{M}], [L_{N}], [L_{O}], [L_{P}], [L_{Q}], [L_{R}], [L_{S}], [L_{T}], [L_{U}], [L_{V}], [L_{W}], [L_{X}], [L_{Y}], [L_{Z}], [L_{?}], [Reflexivity], [Reflexivity_{1}], [Commutativity], [Commutativity_{1}], [<tactic>], [tactic], [[∗ =^{tactic} ∗]], [P(∗, ∗, ∗)], [P^{∗}(∗, ∗, ∗)], [p_{0}], [conclude_{1}(∗, ∗)], [conclude_{2}(∗, ∗, ∗)], [conclude_{3}(∗, ∗, ∗, ∗)], [conclude_{4}(∗, ∗)], [check], [[∗ =^{∘} ∗]], [RootVisible(∗)], [A], [R], [C], [T], [L], [\{∗\}], [\overline{∗}], [a], [b], [c], [d], [e], [f ], [g], [h], [i], [j ], [k], [l], [m], [n], [o], [p], [q], [r], [s], [t], [u], [v], [w], [x], [y], [z], [⟨∗≡ ∗ | ∗:=∗⟩], [⟨∗≡^{0} ∗ | ∗:=∗⟩], [⟨∗≡^{1} ∗ | ∗:=∗⟩], [⟨∗≡^{∗} ∗ | ∗:=∗⟩], [Ded(∗, ∗)], [Ded_{0}(∗, ∗)], [Ded_{1}(∗, ∗, ∗)], [Ded_{2}(∗, ∗, ∗)], [Ded_{3}(∗, ∗, ∗, ∗)], [Ded_{4}(∗, ∗, ∗, ∗)], [Ded_{4}^{∗}(∗, ∗, ∗, ∗)], [Ded_{5}(∗, ∗, ∗)], [Ded_{6}(∗, ∗, ∗, ∗)], [Ded_{6}^{∗}(∗, ∗, ∗, ∗)], [Ded_{7}(∗)], [Ded_{8}(∗, ∗)], [Ded_{8}^{∗}(∗, ∗)], [S], [Neg], [MP], [Gen], [Ded], [S1], [S2], [S3], [S4], [S5], [S6], [S7], [S8], [S9], [Repetition], [A1′], [A2′], [A4′], [A5′], [Prop 3.2a], [Prop 3.2b], [Prop 3.2c], [Prop 3.2d], [Prop 3.2e_{1}], [Prop 3.2e_{2}], [Prop 3.2e], [Prop 3.2f_{1}], [Prop 3.2f_{2}], [Prop 3.2f], [Prop 3.2g_{1}], [Prop 3.2g_{2}], [Prop 3.2g], [Prop 3.2h_{1}], [Prop 3.2h_{2}], [Prop 3.2h], [Block_{1}(∗, ∗, ∗)], [Block_{2}(∗)], [kvanti], [UniqueMember], [UniqueMember(Type)], [SameSeries], [A4], [SameMember], [Qclosed(Addition)], [Qclosed(Multiplication)], [FromCartProd(1)], [1rule fromCartProd(2)], [constantRationalSeries(∗)], [cartProd(∗)], [Power(∗)], [binaryUnion(∗, ∗)], [SetOfRationalSeries], [IsSubset(∗, ∗)], [(p∗, ∗)], [(s∗)], [(⋯)], [Objekt-var], [Ex-var], [Ph-var], [Værdi], [Variabel], [Op(∗)], [Op(∗, ∗)], [∗ ¨ == ∗], [ContainsEmpty(∗)], [Nat(∗)], [Dedu(∗, ∗)], [Dedu_{0}(∗, ∗)], [Dedu_{s}(∗, ∗, ∗)], [Dedu_{1}(∗, ∗, ∗)], [Dedu_{2}(∗, ∗, ∗)], [Dedu_{3}(∗, ∗, ∗, ∗)], [Dedu_{4}(∗, ∗, ∗, ∗)], [Dedu_{4}^{∗}(∗, ∗, ∗, ∗)], [Dedu_{5}(∗, ∗, ∗)], [Dedu_{6}(∗, ∗, ∗, ∗)], [Dedu_{6}^{∗}(∗, ∗, ∗, ∗)], [Dedu_{7}(∗)], [Dedu_{8}(∗, ∗)], [Dedu_{8}^{∗}(∗, ∗)], [Ex_{1}], [Ex_{2}], [Ex3], [Ex_{10}], [Ex_{20}], [∗_{Ex}], [∗^{Ex}], [⟨∗≡ ∗ | ∗:==∗⟩_{Ex}], [⟨∗≡^{0} ∗ | ∗:==∗⟩_{Ex}], [⟨∗≡^{1} ∗ | ∗:==∗⟩_{Ex}], [⟨∗≡^{∗} ∗ | ∗:==∗⟩_{Ex}], [ph_{1}], [ph_{2}], [ph_{3}], [∗_{Ph}], [∗^{Ph}], [⟨∗≡ ∗ | ∗:==∗⟩_{Ph}], [⟨∗≡^{0} ∗ | ∗:==∗⟩_{Ph}], [⟨∗≡^{1} ∗ | ∗:==∗⟩_{Ph}], [⟨∗≡^{∗} ∗ | ∗:==∗⟩_{Ph}], [⟨∗≡ ∗ | ∗:==∗⟩_{Me}], [⟨∗≡^{1} ∗ | ∗:==∗⟩_{Me}], [⟨∗≡^{∗} ∗ | ∗:==∗⟩_{Me}], [bs], [OBS], [BS], [Ø], [SystemQ], [MP], [Gen], [Repetition], [Neg], [Ded], [ExistIntro], [Extensionality], [Ødef], [PairDef], [UnionDef], [PowerDef], [SeparationDef], [AddDoubleNeg], [RemoveDoubleNeg], [AndCommutativity], [AutoImply], [Contrapositive], [FirstConjunct], [SecondConjunct], [FromContradiction], [FromDisjuncts], [IffCommutativity], [IffFirst], [IffSecond], [ImplyTransitivity], [JoinConjuncts], [MP2], [MP3], [MP4], [MP5], [MT], [NegativeMT], [Technicality], [Weakening], [WeakenOr1], [WeakenOr2], [Formula2Pair], [Pair2Formula], [Formula2Union], [Union2Formula], [Formula2Sep], [Sep2Formula], [Formula2Power], [SubsetInPower], [HelperPowerIsSub], [PowerIsSub], [(Switch)HelperPowerIsSub], [(Switch)PowerIsSub], [ToSetEquality], [HelperToSetEquality(t)], [ToSetEquality(t)], [HelperFromSetEquality], [FromSetEquality], [HelperReflexivity], [Reflexivity], [HelperSymmetry], [Symmetry], [HelperTransitivity], [Transitivity], [ERisReflexive], [ERisSymmetric], [ERisTransitive], [ØisSubset], [HelperMemberNotØ], [MemberNotØ], [HelperUniqueØ], [UniqueØ], [==Reflexivity], [==Symmetry], [Helper==Transitivity], [==Transitivity], [HelperTransferNotEq], [TransferNotEq], [HelperPairSubset], [Helper(2)PairSubset], [PairSubset], [SamePair], [SameSingleton], [UnionSubset], [SameUnion], [SeparationSubset], [SameSeparation], [SameBinaryUnion], [IntersectionSubset], [SameIntersection], [AutoMember], [HelperEqSysNotØ], [EqSysNotØ], [HelperEqSubset], [EqSubset], [HelperEqNecessary], [EqNecessary], [HelperNoneEqNecessary], [Helper(2)NoneEqNecessary], [NoneEqNecessary], [EqClassIsSubset], [EqClassesAreDisjoint], [AllDisjoint], [AllDisjointImply], [BSsubset], [Union(BS/R)subset], [UnionIdentity], [EqSysIsPartition], [(x1)], [(x2)], [(y1)], [(y2)], [(v1)], [(v2)], [(v3)], [(v4)], [(v2n)], [(m1)], [(m2)], [(n1)], [(n2)], [(n3)], [(ϵ)], [(ϵ)_{1}], [(ϵ2)], [(fep)], [(fx)], [(fy)], [(fz)], [(fu)], [(fv)], [(fw)], [(rx)], [(ry)], [(rz)], [(ru)], [(sx)], [(sx1)], [(sy)], [(sy1)], [(sz)], [(sz1)], [(su)], [(su1)], [(fxs)], [(fys)], [(crs1)], [(f1)], [(f2)], [(f3)], [(f4)], [(op1)], [(op2)], [(r1)], [(s1)], [(s2)], [X_{1}], [X_{2}], [Y_{1}], [Y_{2}], [V_{1}], [V_{2}], [V_{3}], [V_{4}], [V_{2n}], [M_{1}], [M_{2}], [N_{1}], [N_{2}], [N_{3}], [ϵ], [ϵ1], [ϵ2], [FX], [FY], [FZ], [FU], [FV], [FW], [FEP], [RX], [RY], [RZ], [RU], [(SX)], [(SX1)], [(SY)], [(SY1)], [(SZ)], [(SZ1)], [(SU)], [(SU1)], [FXS], [FYS], [(F1)], [(F2)], [(F3)], [(F4)], [(OP1)], [(OP2)], [(R1)], [(S1)], [(S2)], [(EPob)], [(CRS1ob)], [(F1ob)], [(F2ob)], [(F3ob)], [(F4ob)], [(N1ob)], [(N2ob)], [(OP1ob)], [(OP2ob)], [(R1ob)], [(S1ob)], [(S2ob)], [ph_{4}], [ph_{5}], [ph_{6}], [NAT], [RATIONAL_{S}ERIES], [SERIES], [SetOfReals], [SetOfFxs], [N], [Q], [X], [xs], [xaF], [ysF], [us], [usFoelge], [0], [1], [(−1)], [2], [3], [1/2], [1/3], [2/3], [0f], [1f], [00], [01], [(− − 01)], [02], [01//02], [PlusAssociativity(R)], [PlusAssociativity(R)XX], [Plus0(R)], [Negative(R)], [Times1(R)], [lessAddition(R)], [PlusCommutativity(R)], [LeqAntisymmetry(R)], [LeqTransitivity(R)], [leqAddition(R)], [Distribution(R)], [A4(Axiom)], [InductionAxiom], [EqualityAxiom], [EqLeqAxiom], [EqAdditionAxiom], [EqMultiplicationAxiom], [QisClosed(Reciprocal)(Imply)], [QisClosed(Reciprocal)], [QisClosed(Negative)(Imply)], [QisClosed(Negative)], [leqReflexivity], [leqAntisymmetryAxiom], [leqTransitivityAxiom], [leqTotality], [leqAdditionAxiom], [leqMultiplicationAxiom], [plusAssociativity], [plusCommutativity], [Negative], [plus0], [timesAssociativity], [timesCommutativity], [ReciprocalAxiom], [times1], [Distribution], [0not1], [lemma eqLeq(R)], [TimesAssociativity(R)], [TimesCommutativity(R)], [(Adgic)SameR], [Separation2formula(1)], [Separation2formula(2)], [Cauchy], [PlusF], [ReciprocalF], [From ==], [To ==], [FromInR], [PlusR(Sym)], [ReciprocalR(Axiom)], [LessMinus1(N)], [Nonnegative(N)], [US0], [NextXS(UpperBound)], [NextXS(NoUpperBound)], [NextUS(UpperBound)], [NextUS(NoUpperBound)], [ExpZero], [ExpPositive], [ExpZero(R)], [ExpPositive(R)], [BSzero], [BSpositive], [UStelescope(Zero)], [UStelescope(Positive)], [EqAddition(R)], [FromLimit], [ToUpperBound], [FromUpperBound], [USisUpperBound], [0not1(R)], [ExpUnbounded(R)], [FromLeq(Advanced)(N)], [FromLeastUpperBound], [ToLeastUpperBound], [XSisNotUpperBound], [ysFGreater], [ysFLess], [SmallInverse], [NatType], [RationalType], [SeriesType], [Max], [Numerical], [NumericalF], [MemberOfSeries(Imply)], [JoinConjuncts(2conditions)], [prop lemma imply negation], [TND], [FromNegatedImply], [ToNegatedImply], [FromNegated(2 ∗ Imply)], [FromNegatedAnd], [FromNegatedOr], [ToNegatedOr], [FromNegations], [From3Disjuncts], [From2 ∗ 2Disjuncts], [NegateDisjunct1], [NegateDisjunct2], [ExpandDisjuncts], [SENC1], [SENC2], [LessLeq(R)], [MemberOfSeries], [memberOfSeries(Type)], [sup], [ToNegatedAnd(1)], [UniqueNegative], [DoubleMinus], [MinusNegated], [eqReflexivity], [eqSymmetry], [eqTransitivity], [eqTransitivity4], [eqTransitivity5], [eqTransitivity6], [AddEquations], [SubtractEquations], [SubtractEquationsLeft], [MultiplyEquations], [EqNegated], [PositiveToRight(Eq)], [PositiveToLeft(Eq)(1term)], [NegativeToLeft(Eq)], [NonreciprocalToRight(Eq)(1term)], [PlusAssociativity(4terms)], [LessNeq], [NeqSymmetry], [NeqNegated], [SubNeqRight], [SubNeqLeft], [NegativeToRight(Neq)(1term)], [NeqAddition], [NeqMultiplication], [NonzeroProduct(2)], [UStelescope(+1)], [TelescopeBound(Base)], [TelescopeBound(Indu)], [TelescopeBound], [IntervalSize(Base)], [IntervalSize(Indu)], [IntervalSize], [XS < US], [lemma USdecreasing(+1)], [CloseUS], [CloseUS(n + 1)], [AllNegated(Imply)], [ExistNegated(Imply)], [IntroExist(Helper)], [IntroExist], [ExistMP], [ExistMP2], [TwiceExistMP], [TwiceExistMP2], [EAE − MP], [AddAll], [AddExist(Helper1)], [AddExist(Helper2)], [AddExist], [AddExist(SimpleAnt)], [AddExist(Simple)], [AddEAE], [AEA − negated], [EEA − negated], [Induction], [leqAntisymmetry], [leqTransitivity], [leqAddition], [leqMultiplication], [Reciprocal], [Equality], [eqLeq], [eqAddition], [eqMultiplication], [LeqMultiplicationLeft], [LeqLessEq], [LessLeq], [FromLeqGeq], [subLeqRight], [subLeqLeft], [Leq + 1], [PositiveToRight(Leq)], [PositiveToRight(Leq)(1term)], [lemma negativeToRight(Leq)], [PositiveToLeft(Leq)], [negativeToLeft(Leq)], [negativeToLeft(Leq)(1term)], [LeqAdditionLeft], [leqSubtraction], [leqSubtractionLeft], [thirdGeq], [LeqNegated], [AddEquations(Leq)], [MultiplyEquations(Leq)], [ThirdGeqSeries], [LeqNeqLess], [FromLess], [ToLess], [fromNotLess], [toNotLess], [NegativeLessPositive], [leqLessTransitivity], [LessLeqTransitivity], [LessTransitivity], [LessTotality], [SubLessRight], [SubLessLeft], [SwitchTerms(x < y − z)], [SwitchTerms(x − y < z)], [LessAddition], [LessAdditionLeft], [LessMultiplication], [LessMultiplicationLeft], [LessDivision], [PositiveToRight(Less)], [PositiveToLeft(Less)], [NegativeToLeft(Less)], [NegativeToRight(Less)], [AddEquations(Less)], [AddEquations(LeqLess)], [reciprocalToLeft(Less)], [LessNegated], [PositiveNonzero], [PositiveNegated], [NonpositiveNegated], [NegativeNegated], [NonnegativeNegated], [PositiveHalved], [PositiveInverted], [NonnegativeNumerical], [NegativeNumerical], [PositiveNumerical], [lemma nonpositiveNumerical], [|0| = 0], [0 <= |x|], [x <= |x|], [FromPositiveNumerical], [SameNumerical], [SignNumerical(+)], [SignNumerical], [ToNumericalLess], [FromNumericalGreater], [NumericalDifference], [NumericalDifferenceLess(Helper)], [NumericalDifferenceLess], [SplitNumericalSumHelper], [splitNumericalSum(++)], [splitNumericalSum(−−)], [splitNumericalSum(+ − small)], [splitNumericalSum(+ − big)], [splitNumericalSum(+−)], [splitNumericalSum(−+)], [splitNumericalSum], [SplitNumericalProduct(++)], [SplitNumericalProduct(+−)], [SplitNumericalProduct], [insertMiddleTerm(Numerical)], [insertTwoMiddleTerms(Numerical)], [Three2twoTerms], [Three2threeTerms], [Three2twoFactors], [Three2threeFactors], [Times(−1)], [Times(−1)Left], [MaxLeq(1)], [MaxLeq(2)], [LessThanMax], [x + y = zBackwards], [x ∗ y = zBackwards], [x = x + (y − y)], [x = x + y − y], [x = x ∗ y ∗ (1/y)], [insertMiddleTerm(Sum)], [insertTwoMiddleTerms(Sum)], [insertMiddleTerm(Difference)], [x ∗ 0 + x = x], [x ∗ 0 = 0], [NonnegativeFactors], [NonzeroFactors], [PositiveFactors], [PlusTimesMinus], [MinusTimesMinus], [(−1) ∗ (−1) + (−1) ∗ 1 = 0], [(−1) ∗ (−1) = 1], [0 < 1Helper], [0 < 1], [0 < 2], [0 < 3], [0 < 1/2], [0 < 1/3], [TwoWholes], [ThreeWholes], [TwoHalves], [ThreeThirds], [−x − y = −(x + y)], [−x ∗ y = −(x ∗ y)], [−0 = 0], [SFsymmetry], [SFtransitivity], [f2R(Plus)], [f2R(Times)], [<< TransitivityHelper(Q)], [<< Transitivity], [<<== Reflexivity], [<<== AntisymmetryHelper(Q)], [FromNot < f(Weak)(Helper)], [FromNot < f(Weak)], [FromNot < f(Strong)(Helper2)], [FromNot < f(Strong)(Helper)], [FromNot < f(Strong)], [fromNotSameF(Strongest)(Helper2)], [fromNotSameF(Strongest)(Helper)], [fromNotSameF(Strongest)], [ToLess(F)(Helper)], [ToLess(F)], [FromNot <<], [ToLess(R)], [FromNotSameF(Weak)(Helper)], [FromNotSameF(Weak)], [FromNotLess(F)], [== Addition], [== AdditionLeft], [Fpart − Bounded(Base)], [Fpart − Bounded(InduHelper)], [Fpart − Bounded(Indu)], [Fpart − Bounded], [F − Bounded(Helper)], [F − Bounded], [SameFmultiplication(Helper)], [SameFmultiplication], [EqMultiplication(R)], [EqMultiplicationLeft(R)], [x ∗ 0 = 0(F)], [x ∗ 0 = 0(R)], [LessMultiplication(F)(Helper2)], [LessMultiplication(F)(Helper)], [LessMultiplication(F)], [LessMultiplication(R)], [LeqMultiplication(R)], [PlusAssociativity(F)], [Plus0(F)], [PlusCommutativity(F)], [TimesAssociativity(F)], [Times1f], [Cauchy(2)(Helper)], [Cauchy(2)], [ReciprocalFnonzero], [(Eventually = f)2sameF(Helper)], [(Eventually = f)2sameF], [FromNotSameF(Strong)(Helper2)], [FromNotSameF(Strong)(Helper)], [FromNotSameF(Strong)], [SameFreciprocal(Helper)], [SameFreciprocal], [From!! ==], [Reciprocal(R)], [TimesCommutativity(F)], [Distribution(F)], [FromMax(1)], [FromMax(2)], [ToNegatedAnd], [DistributionOut], [DistributionOutLeft], [DistributionLeft], [FromNotLess(R)], [CartProdIsRelation], [FromSubset], [SubsetIsRelation], [ToSeries], [FromSeries], [SeriesSubsetCP], [ValueType], [RemoveOr], [FromSingleton], [InPair(1)], [InPair(2)], [SameMember(2)], [ToBinaryUnion(1)], [ToBinaryUnion(2)], [FromOrderedPair(TwoLevels)], [ToCartProd(Helper)], [ToCartProd], [NonreciprocalToRight(Eq)], [NonreciprocalToLeft(Eq)(1term)], [SameReciprocal], [CPseparationIsRelation], [OrderedPairEquality], [ReciprocalIsFunction], [ReciprocalIsTotal], [ReciprocalIsRationalSeries], [CrsIsRelation], [CrsIsFunction], [CrsIsTotal], [CrsIsSeries], [CrsLookup], [0f], [1f], [ToSingleton], [FromSameSingleton], [SingletonmembersEqual], [UnequalsNotInSingleton], [NonsingletonmembersUnequal], [FromOrderedPair], [FromOrderedPair(1)], [FromOrderedPair(2)], [FromCartProd], [FromCartProd(1)], [FromCartProd(2)], [sameOrderedPair], [InSeriesHelper], [InSeries], [To = f(Subset)(Helper)], [To = f(Subset)], [To = f], [productIsFunction], [productIsTotal], [ProductIsRationalSeries], [TimesF], [−x + (1/2)x = −(1/2)x], [PositiveTripled], [PositiveDividedBy3], [|x − x| = 0], [1 < 2], [1/3 < 2/3], [(1/3)x + (1/3)x = (2/3)x], [(2/3)x + (1/3)x = x], [−x + (2/3)x = −(1/3)x], [−(1/3)x − (1/3)x = −(2/3)x], [−x + (1/3)x = −(2/3)x], [PreserveLessGreater], [ClosetolessIsLess], [SubLessLeft(F)], [SubLessLeft(R)], [ClosetogreaterIsGreater], [SubLessRight(F)], [SubLessRight(R)], [plus0Left], [times1Left], [EqAdditionLeft], [EqMultiplicationLeft], [PlusF(Sym)], [TimesF(Sym)], [SameSeries(Gen)], [EqualsSameF], [LeqReflexivity(R)], [LeqTotality(R)], [PositiveToLeft(Eq)], [ExpZero(Exact)], [SameExp(Base)], [SameExp(Indu)], [SameExp], [(1/2)(x + y) − x = (1/2)(y − x)], [y − (1/2)(x + y) = (1/2)(y − x)], [BSzero(Exact)], [SameBS(2)(Base)], [SameBS(2)(Indu)], [SameBS(2)], [NegativeToLeft(Less)(1term)], [BS(+1)], [BSbound(Exact)(Base)], [BSbound(Exact)(Indu)], [BSbound(Exact)], [BSbound], [UStelescope(Zero)(Exact)], [SameTelescope(2)(Base)], [SameTelescope(2)(Indu)], [SameTelescope(2)], [Exp(+1)], [PositiveBase(Base)], [PositiveBase(Indu)], [PositiveBase], [TelescopeNumerical(Base)], [TelescopeNumerical(Indu)], [TelescopeNumerical], [(+1)IsPositive(N)], [DistributionOut(Minus)], [PositiveToRight(Eq)(1term)], [SameSeries(NumDiff)], [ToNegatedDoubleImply], [AddNegatedAll], [(A)to( E )(Imply)], [(E)to( A )(Imply)], [(E )to( A)(Imply)], [ToNegatedAEA], [Three2threeTerms(R)], [LessNeq(F)(Helper)], [LessNeq(F)], [LessNeq(R)], [x = x + (y − y)(R)], [x = x + y − y(R)], [SubtractEquations(R)], [NeqAddition(R)], [PositiveToRight(Less)(R)], [PositiveToRight(Less)(1term)(R)], [LeqNeqLess(R)], [SubLeqLeft(R)], [ToLeq(Advanced)(R)], [LeqLessTransitivity(R)], [NegativeToLeft(Eq)(R)], [NegativeToRight(Less)(R)], [!! == Symmetry], [SwitchTerms(x <= y − z)], [Plus0Left(R)], [PositiveToRight(Eq)(R)], [EqAdditionLeft(R)], [Three2twoTerms(R)], [To!! ==], [PositiveToRight(Less)(1term)], [(A )to( E)], [NegativeToRight(Eq)(R)], [NegativeToRight(Eq)(1term)(R)], [DoubleMinus(R)], [UniqueNegative(R)], [SubtractEquationsLeft(R)], [EqNegated(R)], [NeqNegated(R)], [−0 = 0(R)], [NegativeNegated(R)], [FromLeqGeq(R)], [0 <= |x|(R)], [PositiveNegated(R)], [AddEquations(R)], [Times(−1)(R)], [Times(−1)Left(R)], [−x − y = −(x + y)(R)], [LessTotality(R)], [SameNumerical(R)], [MinusNegated(R)], [PositiveNumerical(R)], [SignNumerical(+)(R)], [NonnegativeNumerical(R)], [NegativeNumerical(R)], [LeqNegated(R)], [LessNegated(R)], [SubLeqRight(R)], [FromLess(R)], [DistributionOut(R)], [x ∗ 0 + x = x(R)], [x ∗ 0 = 0(R)(fff)], [SignNumerical(R)], [NumericalDifference(R)], [x <= |x|(R)], [USlimitIsUpperBound(Helper)], [USlimitIsUpperBound], [(−1) ∗ (−1) + (−1) ∗ 1 = 0(R)], [(−1) ∗ (−1) = 1(R)], [0 < 1Helper(R)], [0 < 1(R)], [ExpZero(Exact)(R)], [PositiveBase(R)(Base)], [Three2twoFactors(R)], [x = x ∗ y ∗ (1/y)(R)], [NeqMultiplication(R)], [LessTransitivity(R)], [0 < 2(R)], [SameExp(R)(Base)], [SameExp(R)(Indu)], [SameExp(R)], [SubNeqLeft(R)], [SubNeqRight(R)], [NonzeroFactors(R)], [NonnegativeFactors(R)], [PositiveFactors(R)], [LessDivision(R)], [0 < 1/2(R)], [PositiveToRight(Eq)(1term)(R)], [Exp(+1)(R)], [PositiveBase(R)(Indu)], [PositiveBase(R)], [−x ∗ y = −(x ∗ y)(R)], [PositiveToLeft(Eq)(R)], [Times1Left(R)], [x + x = 2 ∗ x(R)], [(1/2)x + (1/2)x = x(R)], [DistributionOut(Minus)(R)], [(1/2)(x + y) − x = (1/2)(y − x)(R)], [IntervalSize(R)(Base)], [LessMultiplicationLeft(R)], [NegativeToLeft(Less)(R)], [NegativeToLeft(Less)(1term)(R)], [y − (1/2)(x + y) = (1/2)(y − x)(R)], [IntervalSize(R)(Indu)], [IntervalSize(R)], [XSlessUS(R)], [USdecreasing(+1)(R)], [ExpUnbounded(Base)], [ExpUnbounded(Indu)], [ExpUnbounded], [1 <= x + 1(N)], [ExpNonzero(Base)], [ExpNonzero(Indu)], [ExpNonzero], [ExpNonzero(2)], [HalfBase(Base)], [HalfBase(Indu)], [MultiplyEquations(R)], [NonreciprocalToRight(Eq)(1term)(R)], [PositiveNonzero(R)], [NonzeroProduct(2)(R)], [HalfBase], [Three2threeFactors(R)], [x ∗ y = zBackwards(R)], [PositiveInverted(R)], [ReciprocalToRight(Less)(R)], [ReciprocalToRight(Less)(1term)(R)], [NonreciprocalToLeft(Less)(R)], [1 < x ∗ y(R)], [SwitchFactors(1/x < y)(R)], [SmallHalving], [IntervalSize(anyPositive)], [USdecreasing(+n)(Base)], [USdecreasing(+n)(Indu)], [USdecreasing(+n)], [USdecreasing], [LeqAdditionLeft(R)], [ToNotLess(R)], [LimitOfUSIsLeq], [SubtractEquations(Less)(R)], [SubtractEquationsLeft(Less)(R)], [LessNegated(Negative)(R)], [FromNegatedAnd(Imply)], [RemoveDoubleNeg(Consequent)], [FromNotUpperBound], [LeqNUB], [USlimitIsLeastUpperBound(Helper)], [USlimitIsLeastUpperBound], [ExistMP3], [GreaterPositive(N)], [ysFClose(Helper)], [ysFClose], [ysFCauchy(Helper)], [ysFCauchy], [from <<==], [to <<==], [NonnegativeNumerical(F)], [NegativeNumerical(F)]; Preassociative [tester1], [tester2], [tester3], [tester4], [tester5], [tester6]; Preassociative [∗ {∗}], [∗/indexintro(∗, ∗, ∗, ∗)], [∗/intro(∗, ∗, ∗)], [∗/bothintro(∗, ∗, ∗, ∗, ∗)], [∗/nameintro(∗, ∗, ∗, ∗)], [∗′], [∗[ ∗]], [∗[∗→∗]], [∗[∗⇒∗]], [∗0], [∗1], [0b], [∗-color(∗)], [∗-color^{∗}(∗)], [∗^{H}], [∗^{T}], [∗^{U}], [∗^{h}], [∗^{t}], [∗^{s}], [∗^{c}], [∗^{d}], [∗^{a}], [∗^{C}], [∗^{M}], [∗^{B}], [∗^{r}], [∗^{i}], [∗^{d}], [∗^{R}], [∗^{0}], [∗^{1}], [∗^{2}], [∗^{3}], [∗^{4}], [∗^{5}], [∗^{6}], [∗^{7}], [∗^{8}], [∗^{9}], [∗^{E}], [∗^{V}], [∗^{C}], [∗^{C^{∗}}], [∗^{hide}]; Preassociative [“ ∗ ”], [], [(∗)^{t}], [string(∗) + ∗], [string(∗) ++ ∗], [ ∗], [ ∗], [!∗], ["∗], [#∗], [$∗], [%∗], [&∗], [’∗], [(∗], [)∗], [∗∗], [+∗], [, ∗], [-∗], [.∗], [/∗], [0∗], [1∗], [2∗], [3∗], [4∗], [5∗], [6∗], [7∗], [8∗], [9∗], [:∗], [; ∗], [<∗], [=∗], [>∗], [?∗], [@∗], [A∗], [B∗], [C∗], [D∗], [E∗], [F∗], [G∗], [H∗], [I∗], [J∗], [K∗], [L∗], [M∗], [N∗], [O∗], [P∗], [Q∗], [R∗], [S∗], [T∗], [U∗], [V∗], [W∗], [X∗], [Y∗], [Z∗], [[∗], [\∗], []∗], [ˆ ∗], [ ∗], [‘∗], [a∗], [b∗], [c∗], [d∗], [e∗], [f∗], [g∗], [h∗], [i∗], [j∗], [k∗], [l∗], [m∗], [n∗], [o∗], [p∗], [q∗], [r∗], [s∗], [t∗], [u∗], [v∗], [w∗], [x∗], [y∗], [z∗], [{∗], [|∗], [}∗], [˜ ∗], [Preassociative ∗; ∗], [Postassociative ∗; ∗], [[∗], ∗], [priority ∗ end], [newline ∗], [macro newline ∗], [MacroIndent(∗)]; Preassociative [∗ ’ ∗], [∗ ‘ ∗]; Preassociative [∗(exp)∗]; Preassociative [∗′], [R(∗)], [− − R(∗)], [rec∗]; Preassociative [∗/∗], [∗ ∩ ∗], [∗[∗]]; Preassociative [∪∗], [∗ ∪ ∗], [P(∗)]; Preassociative [{∗}], [StateExpand(∗, ∗, ∗)], [extractSeries(∗)], [SetOfSeries(∗)], [− − Macro(∗)], [ExpandList(∗, ∗, ∗)], [∗ ∗ Macro(∗)], [+ + Macro(∗)], [<< Macro(∗)], [||Macro(∗)], [01//Macro(∗)], [UB(∗, ∗)], [LUB(∗, ∗)], [BS(∗, ∗)], [UStelescope(∗, ∗)], [(∗)], [|f ∗ |], [|r ∗ |], [Limit(∗, ∗)], [Union(∗)], [IsOrderedPair(∗, ∗, ∗)], [IsRelation(∗, ∗, ∗)], [isFunction(∗, ∗, ∗)], [IsSeries(∗, ∗)], [IsNatural(∗, ∗)], [OrderedPair(∗, ∗)], [TypeNat(∗)], [TypeNat0(∗)], [TypeRational(∗)], [TypeRational0(∗)], [TypeSeries(∗, ∗)], [Typeseries0(∗, ∗)]; Preassociative [{∗, ∗}], [⟨∗, ∗⟩], [(−u∗)], [−_{f}∗], [(− − ∗)], [1f/∗], [01//temp∗]; Preassociative [∗(∗, ∗)], [ReflRel(∗, ∗)], [SymRel(∗, ∗)], [TransRel(∗, ∗)], [EqRel(∗, ∗)], [[∗ ∈ ∗]_{∗}], [Partition(∗, ∗)]; Preassociative [∗ · ∗], [∗ ·_{0} ∗], [(∗ ∗ ∗)], [∗ ∗_{f} ∗], [∗ ∗ ∗∗]; Preassociative [∗ + ∗], [∗ +_{0} ∗], [∗ +_{1} ∗], [∗ − ∗], [∗ −_{0} ∗], [∗ −_{1} ∗], [(∗ + ∗)], [(∗ − ∗)], [∗ +_{f} ∗], [∗ −_{f} ∗], [∗ + +∗], [R(∗) − −R(∗)]; Preassociative [∗ ∈ ∗]; Preassociative [| ∗ |], [if(∗, ∗, ∗)], [Max(∗, ∗)], [Max(∗, ∗)]; Preassociative [∗ = ∗], [∗ ≠ ∗], [∗ <= ∗], [∗ < ∗], [∗ <_{f} ∗], [∗ ≤_{f} ∗], [SF(∗, ∗)], [∗ == ∗], [∗!! == ∗], [∗ << ∗], [∗ <<== ∗]; Preassociative [∗ ∪ {∗}], [∗ ∪ ∗], [∗\{∗}]; Postassociative [∗ ˙. . ∗], [∗ \underline{˙. .} ∗], [∗ \underline{: :} ∗], [∗ \underline{+2∗} ∗], [∗ : : ∗], [∗ +2∗ ∗]; Postassociative [∗, ∗]; Preassociative

[∗ ≈^{B} ∗], [∗ ≈^{D} ∗], [∗ ≈^{C} ∗], [∗ ≈^{P} ∗], [∗ ≈ ∗], [∗ = ∗], [∗ →^{+} ∗], [∗ =^{t} ∗], [∗ =^{t^{∗}} ∗], [∗ =^{r} ∗],

[∗ ∈_{t} ∗], [∗ ⊆_{T} ∗], [∗ =^{T} ∗], [∗ =^{s} ∗], [∗ free in ∗], [∗ free in^{∗} ∗], [∗ free for ∗ in ∗], [∗ free for^{∗} ∗ in ∗], [∗ ∈_{c} ∗], [∗ < ∗], [∗ <′ ∗], [∗ ≤′ ∗], [∗ = ∗], [∗ ≠ ∗], [∗^{var}], [∗#^{0}∗], [∗#^{1}∗], [∗#^{∗}∗], [∗==∗], [∗ ⊆ ∗]; Preassociative [¬∗], [ ˙¬ (∗)n], [∗ ∉ ∗], [∗ ≠ ∗]; Preassociative [∗ ∧ ∗], [∗ ¨∧ ∗], [∗ ˜∧ ∗], [∗ ∧_{c} ∗], [∗ ˙∧ ∗]; Preassociative [∗ ∨ ∗], [∗ ∥ ∗], [∗ ¨∨ ∗]; Postassociative [∗ ˙∨ ∗]; Preassociative [∃∗: ∗], [∀∗: ∗], [∀_{obj}∗: ∗], [∃∗: ∗]; Postassociative [∗ ¨⇒ ∗], [∗ ⇒ ∗], [∗ ⇔ ∗], [∗ ˙⇔ ∗]; Preassociative [{ph ∈ ∗ | ∗}]; Postassociative [∗ : ∗], [∗ spy ∗], [∗!∗]; Preassociative

[∗ \{ \begin{matrix}∗ \\ ∗ \end{matrix} ];

Preassociative [λ ∗ .∗], [Λ ∗ .∗], [Λ∗], [if ∗ then ∗ else ∗], [let ∗ = ∗ in ∗], [let ∗ ¨= ∗ in ∗]; Preassociative [∗#∗]; Preassociative [∗^{I}], [∗^{⊳} ], [∗^{V}], [∗^{+}], [∗^{−}], [∗^{∗}]; Preassociative [∗ @ ∗], [∗ ⊳ ∗], [∗ ⊳ ⊳ ∗], [∗ ≫ ∗], [∗ ⊵ ∗]; Postassociative [∗ ⊢ ∗], [∗ ⊢⊢ ∗], [∗ i.e. ∗]; Preassociative [∀∗: ∗], [Π∗: ∗]; Postassociative [∗ ⊕ ∗]; Postassociative [∗; ∗]; Preassociative [∗ proves ∗]; Preassociative [∗ proof of ∗ : ∗], [Line ∗ : ∗ ≫ ∗; ∗], [Last line ∗ ≫ ∗ □], [Line ∗ : Premise ≫ ∗; ∗], [Line ∗ : Side-condition ≫ ∗; ∗], [Arbitrary ≫ ∗; ∗], [Local ≫ ∗ = ∗; ∗], [Begin ∗; ∗ : End; ∗], [Last block line ∗ ≫ ∗; ], [Arbitrary ≫ ∗; ∗]; Postassociative [∗ | ∗]; Postassociative [∗ , ∗], [∗[ ∗]∗]; Preassociative [∗&∗]; Preassociative [∗\\∗], [∗ linebreak[4] ∗], [∗\\∗]; ]

[sup2 →^{tex} “sup2”]

[sup2 →^{pyk} “sup2”]

LeqTotality(R)

[LeqTotality(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⊢ ToLess(R) ⊳ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ≫ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n; LessLeq(R) ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; ∀\underline{(fx)}: ∀\underline{(fy)}: Ded ⊳ ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; Repetition ⊳ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, p_{0}, c)] [LeqTotality(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[LeqTotality(R) →^{tex} “LeqTotality(R)”]

[LeqTotality(R) →^{pyk} “lemma leqTotality(R)”]

PositiveToLeft(Eq)

[PositiveToLeft(Eq) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = (\underline{y} + \underline{z}) ⊢ eqAddition ⊳ \underline{x} = (\underline{y} + \underline{z}) ≫ (\underline{x} + (−u\underline{z})) = ((\underline{y} + \underline{z}) + (−u\underline{z})); x = x + y − y ≫ \underline{y} = ((\underline{y} + \underline{z}) + (−u\underline{z})); eqSymmetry ⊳ \underline{y} = ((\underline{y} + \underline{z}) + (−u\underline{z})) ≫ ((\underline{y} + \underline{z}) + (−u\underline{z})) = \underline{y}; eqTransitivity ⊳ (\underline{x} + (−u\underline{z})) = ((\underline{y} + \underline{z}) + (−u\underline{z})) ⊳ ((\underline{y} + \underline{z}) + (−u\underline{z})) = \underline{y} ≫ (\underline{x} + (−u\underline{z})) = \underline{y}⌉, p_{0}, c)]

[PositiveToLeft(Eq) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = (\underline{y} + \underline{z}) ⊢ (\underline{x} + (−u\underline{z})) = \underline{y}]

[PositiveToLeft(Eq) →^{tex} “PositiveToLeft(Eq)”]

[PositiveToLeft(Eq) →^{pyk} “lemma positiveToLeft(Eq)”]

ExpZero(Exact)

[ExpZero(Exact) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: eqReflexivity ≫ 0 = 0; ExpZero ⊳ 0 = 0 ≫ \underline{x}(exp)0 = 1⌉, p_{0}, c)]

[ExpZero(Exact) →^{stmt} SystemQ ⊢ ∀\underline{x}: \underline{x}(exp)0 = 1]

[ExpZero(Exact) →^{tex} “ExpZero(Exact) ”]

[ExpZero(Exact) →^{pyk} “lemma expZero exact”]

SameExp(Base)

[SameExp(Base) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{n}: ∀\underline{x}: 0 = \underline{n} ⊢ ExpZero(Exact) ≫ \underline{x}(exp)0 = 1; eqSymmetry ⊳ 0 = \underline{n} ≫ \underline{n} = 0; ExpZero ⊳ \underline{n} = 0 ≫ \underline{x}(exp)\underline{n} = 1; eqSymmetry ⊳ \underline{x}(exp)\underline{n} = 1 ≫ 1 = \underline{x}(exp)\underline{n}; eqTransitivity ⊳ \underline{x}(exp)0 = 1 ⊳ 1 = \underline{x}(exp)\underline{n} ≫ \underline{x}(exp)0 = \underline{x}(exp)\underline{n}; ∀\underline{n}: ∀\underline{x}: Ded ⊳ ∀\underline{n}: ∀\underline{x}: 0 = \underline{n} ⊢ \underline{x}(exp)0 = \underline{x}(exp)\underline{n} ≫ 0 = \underline{n} ⇒ \underline{x}(exp)0 = \underline{x}(exp)\underline{n}; Gen ⊳ 0 = \underline{n} ⇒ \underline{x}(exp)0 = \underline{x}(exp)\underline{n} ≫ ∀_{obj}\underline{n}: 0 = \underline{n} ⇒ \underline{x}(exp)0 = \underline{x}(exp)\underline{n}⌉, p_{0}, c)]

[SameExp(Base) →^{stmt} SystemQ ⊢ ∀\underline{n}: ∀\underline{x}: ∀_{obj}\underline{n}: 0 = \underline{n} ⇒ \underline{x}(exp)0 = \underline{x}(exp)\underline{n}]

[SameExp(Base) →^{tex} “SameExp(Base)”]

[SameExp(Base) →^{pyk} “lemma sameExp base”]

SameExp(Indu)

[SameExp(Indu) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: ∀\underline{x}: ∀\underline{m}: ∀\underline{n}: ∀\underline{x}: ∀_{obj}\underline{n}: \underline{m} = \underline{n} ⇒ \underline{x}(exp)\underline{m} = \underline{x}(exp)\underline{n} ⊢ (\underline{m} + 1) = \underline{n} ⊢ (+1)IsPositive(N) ≫ ˙¬ (0 <= (\underline{m} + 1) ⇒ ˙¬ ( ˙¬ (0 = (\underline{m} + 1))n)n)n; ExpPositive ⊳ ˙¬ (0 <= (\underline{m} + 1) ⇒ ˙¬ ( ˙¬ (0 = (\underline{m} + 1))n)n)n ≫ \underline{x}(exp)(\underline{m} + 1) = (\underline{x} ∗ \underline{x}(exp)((\underline{m} + 1) + (−u1))); x = x + y − y ≫ \underline{m} = ((\underline{m} + 1) + (−u1)); A4 @((\underline{m} + 1) + (−u1)) ⊳ ∀_{obj}\underline{n}: \underline{m} = \underline{n} ⇒ \underline{x}(exp)\underline{m} = \underline{x}(exp)\underline{n} ≫ \underline{m} = ((\underline{m} + 1) + (−u1)) ⇒ \underline{x}(exp)\underline{m} = \underline{x}(exp)((\underline{m} + 1) + (−u1)); MP ⊳ \underline{m} = ((\underline{m} + 1) + (−u1)) ⇒ \underline{x}(exp)\underline{m} = \underline{x}(exp)((\underline{m} + 1) + (−u1)) ⊳ \underline{m} = ((\underline{m} + 1) + (−u1)) ≫ \underline{x}(exp)\underline{m} = \underline{x}(exp)((\underline{m} + 1) + (−u1)); eqSymmetry ⊳ \underline{x}(exp)\underline{m} = \underline{x}(exp)((\underline{m} + 1) + (−u1)) ≫ \underline{x}(exp)((\underline{m} + 1) + (−u1)) = \underline{x}(exp)\underline{m}; EqMultiplicationLeft ⊳ \underline{x}(exp)((\underline{m} + 1) + (−u1)) = \underline{x}(exp)\underline{m} ≫ (\underline{x} ∗ \underline{x}(exp)((\underline{m} + 1) + (−u1))) = (\underline{x} ∗ \underline{x}(exp)\underline{m}); A4 @(\underline{n} + (−u1)) ⊳ ∀_{obj}\underline{n}: \underline{m} = \underline{n} ⇒ \underline{x}(exp)\underline{m} = \underline{x}(exp)\underline{n} ≫ \underline{m} = (\underline{n} + (−u1)) ⇒ \underline{x}(exp)\underline{m} = \underline{x}(exp)(\underline{n} + (−u1)); PositiveToRight(Eq) ⊳ (\underline{m} + 1) = \underline{n} ≫ \underline{m} = (\underline{n} + (−u1)); MP ⊳ \underline{m} = (\underline{n} + (−u1)) ⇒ \underline{x}(exp)\underline{m} = \underline{x}(exp)(\underline{n} + (−u1)) ⊳ \underline{m} = (\underline{n} + (−u1)) ≫ \underline{x}(exp)\underline{m} = \underline{x}(exp)(\underline{n} + (−u1)); EqMultiplicationLeft ⊳ \underline{x}(exp)\underline{m} = \underline{x}(exp)(\underline{n} + (−u1)) ≫ (\underline{x} ∗ \underline{x}(exp)\underline{m}) = (\underline{x} ∗ \underline{x}(exp)(\underline{n} + (−u1))); SubLessRight ⊳ (\underline{m} + 1) = \underline{n} ⊳ ˙¬ (0 <= (\underline{m} + 1) ⇒ ˙¬ ( ˙¬ (0 = (\underline{m} + 1))n)n)n ≫ ˙¬ (0 <= \underline{n} ⇒ ˙¬ ( ˙¬ (0 = \underline{n})n)n)n; ExpPositive ⊳ ˙¬ (0 <= \underline{n} ⇒ ˙¬ ( ˙¬ (0 = \underline{n})n)n)n ≫ \underline{x}(exp)\underline{n} = (\underline{x} ∗ \underline{x}(exp)(\underline{n} + (−u1))); eqSymmetry ⊳ \underline{x}(exp)\underline{n} = (\underline{x} ∗ \underline{x}(exp)(\underline{n} + (−u1))) ≫ (\underline{x} ∗ \underline{x}(exp)(\underline{n} + (−u1))) = \underline{x}(exp)\underline{n}; eqTransitivity5 ⊳ \underline{x}(exp)(\underline{m} + 1) = (\underline{x} ∗ \underline{x}(exp)((\underline{m} + 1) + (−u1))) ⊳ (\underline{x} ∗ \underline{x}(exp)((\underline{m} + 1) + (−u1))) = (\underline{x} ∗ \underline{x}(exp)\underline{m}) ⊳ (\underline{x} ∗ \underline{x}(exp)\underline{m}) = (\underline{x} ∗ \underline{x}(exp)(\underline{n} + (−u1))) ⊳ (\underline{x} ∗ \underline{x}(exp)(\underline{n} + (−u1))) = \underline{x}(exp)\underline{n} ≫ \underline{x}(exp)(\underline{m} + 1) = \underline{x}(exp)\underline{n}; Ded ⊳ ∀\underline{m}: ∀\underline{n}: ∀\underline{x}: ∀_{obj}\underline{n}: \underline{m} = \underline{n} ⇒ \underline{x}(exp)\underline{m} = \underline{x}(exp)\underline{n} ⊢ (\underline{m}+1) = \underline{n} ⊢ \underline{x}(exp)(\underline{m}+1) = \underline{x}(exp)\underline{n} ≫ ∀_{obj}\underline{n}: \underline{m} = \underline{n} ⇒ \underline{x}(exp)\underline{m} = \underline{x}(exp)\underline{n} ⇒ (\underline{m} + 1) = \underline{n} ⇒ \underline{x}(exp)(\underline{m} + 1) = \underline{x}(exp)\underline{n}; ∀_{obj}\underline{n}: \underline{m} = \underline{n} ⇒ \underline{x}(exp)\underline{m} = \underline{x}(exp)\underline{n} ⊢ MP ⊳ ∀_{obj}\underline{n}: \underline{m} = \underline{n} ⇒ \underline{x}(exp)\underline{m} = \underline{x}(exp)\underline{n} ⇒ (\underline{m} + 1) = \underline{n} ⇒ \underline{x}(exp)(\underline{m} + 1) = \underline{x}(exp)\underline{n} ⊳ ∀_{obj}\underline{n}: \underline{m} = \underline{n} ⇒ \underline{x}(exp)\underline{m} = \underline{x}(exp)\underline{n} ≫ (\underline{m} + 1) = \underline{n} ⇒ \underline{x}(exp)(\underline{m} + 1) = \underline{x}(exp)\underline{n}; Gen ⊳ (\underline{m} + 1) = \underline{n} ⇒ \underline{x}(exp)(\underline{m} + 1) = \underline{x}(exp)\underline{n} ≫ ∀_{obj}\underline{n}: (\underline{m} + 1) = \underline{n} ⇒ \underline{x}(exp)(\underline{m}+1) = \underline{x}(exp)\underline{n}; ∀\underline{m}: ∀\underline{n}: ∀\underline{x}: Ded⊳∀\underline{m}: ∀\underline{n}: ∀\underline{x}: ∀_{obj}\underline{n}: \underline{m} = \underline{n} ⇒ \underline{x}(exp)\underline{m} = \underline{x}(exp)\underline{n} ⊢ ∀_{obj}\underline{n}: (\underline{m} + 1) = \underline{n} ⇒ \underline{x}(exp)(\underline{m} + 1) = \underline{x}(exp)\underline{n} ≫ ∀_{obj}\underline{n}: \underline{m} = \underline{n} ⇒ \underline{x}(exp)\underline{m} = \underline{x}(exp)\underline{n} ⇒ ∀_{obj}\underline{n}: (\underline{m} + 1) = \underline{n} ⇒ \underline{x}(exp)(\underline{m} + 1) = \underline{x}(exp)\underline{n}⌉, p_{0}, c)]

[SameExp(Indu) →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: ∀\underline{x}: ∀_{obj}\underline{n}: \underline{m} = \underline{n} ⇒ \underline{x}(exp)\underline{m} = \underline{x}(exp)\underline{n} ⇒ ∀_{obj}\underline{n}: (\underline{m} + 1) = \underline{n} ⇒ \underline{x}(exp)(\underline{m} + 1) = \underline{x}(exp)\underline{n}]

[SameExp(Indu) →^{tex} “SameExp(Indu)”]

[SameExp(Indu) →^{pyk} “lemma sameExp indu”]

SameExp

[SameExp →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: ∀\underline{x}: \underline{m} = \underline{n} ⊢ SameExp(Base) ≫ ∀_{obj}\underline{n}: 0 = \underline{n} ⇒ \underline{x}(exp)0 = \underline{x}(exp)\underline{n}; SameExp(Indu) ≫ ∀_{obj}\underline{n}: \underline{m} = \underline{n} ⇒ \underline{x}(exp)\underline{m} = \underline{x}(exp)\underline{n} ⇒ ∀_{obj}\underline{n}: (\underline{m} + 1) = \underline{n} ⇒ \underline{x}(exp)(\underline{m} + 1) = \underline{x}(exp)\underline{n}; Induction ⊳ ∀_{obj}\underline{n}: 0 = \underline{n} ⇒ \underline{x}(exp)0 = \underline{x}(exp)\underline{n} ⊳ ∀_{obj}\underline{n}: \underline{m} = \underline{n} ⇒ \underline{x}(exp)\underline{m} = \underline{x}(exp)\underline{n} ⇒ ∀_{obj}\underline{n}: (\underline{m} + 1) = \underline{n} ⇒ \underline{x}(exp)(\underline{m} + 1) = \underline{x}(exp)\underline{n} ≫ ∀_{obj}\underline{n}: \underline{m} = \underline{n} ⇒ \underline{x}(exp)\underline{m} = \underline{x}(exp)\underline{n}; A4 @ \underline{n} ⊳ ∀_{obj}\underline{n}: \underline{m} = \underline{n} ⇒ \underline{x}(exp)\underline{m} = \underline{x}(exp)\underline{n} ≫ \underline{m} = \underline{n} ⇒ \underline{x}(exp)\underline{m} = \underline{x}(exp)\underline{n}; MP ⊳ \underline{m} = \underline{n} ⇒ \underline{x}(exp)\underline{m} = \underline{x}(exp)\underline{n} ⊳ \underline{m} = \underline{n} ≫ \underline{x}(exp)\underline{m} = \underline{x}(exp)\underline{n}⌉, p_{0}, c)]

[SameExp →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: ∀\underline{x}: \underline{m} = \underline{n} ⊢ \underline{x}(exp)\underline{m} = \underline{x}(exp)\underline{n}]

[SameExp →^{tex} “SameExp”]

[SameExp →^{pyk} “lemma sameExp”]

(1/2)(x + y) − x = (1/2)(y − x)

[(1/2)(x+y)−x = (1/2)(y−x) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: Distribution ≫ (rec(1+1)∗(\underline{x}+\underline{y})) = ((rec(1+1)∗\underline{x})+(rec(1+1)∗\underline{y})); eqAddition⊳(rec(1+1)∗ (\underline{x} + \underline{y})) = ((rec(1 + 1) ∗ \underline{x}) + (rec(1 + 1) ∗ \underline{y})) ≫ ((rec(1 + 1) ∗ (\underline{x} + \underline{y})) + (−u\underline{x})) = (((rec(1 + 1) ∗ \underline{x}) + (rec(1 + 1) ∗ \underline{y})) + (−u\underline{x})); plusCommutativity ≫ ((rec(1 + 1) ∗ \underline{x}) + (rec(1 + 1) ∗ \underline{y})) = ((rec(1+1)∗\underline{y})+(rec(1+1)∗\underline{x})); eqAddition⊳((rec(1+1)∗\underline{x})+(rec(1+1)∗\underline{y})) = ((rec(1+1)∗\underline{y})+(rec(1+1)∗\underline{x})) ≫ (((rec(1+1)∗\underline{x})+(rec(1+1)∗\underline{y}))+(−u\underline{x})) = (((rec(1 + 1) ∗ \underline{y}) + (rec(1 + 1) ∗ \underline{x})) + (−u\underline{x})); plusAssociativity ≫ (((rec(1 + 1) ∗ \underline{y}) + (rec(1 + 1) ∗ \underline{x})) + (−u\underline{x})) = ((rec(1 + 1) ∗ \underline{y}) + ((rec(1 + 1) ∗ \underline{x}) + (−u\underline{x}))); TwoHalves ≫ ((rec(1 + 1) ∗ \underline{x}) + (rec(1 + 1) ∗ \underline{x})) = \underline{x}; PositiveToRight(Eq) ⊳ ((rec(1 + 1) ∗ \underline{x}) + (rec(1 + 1) ∗ \underline{x})) = \underline{x} ≫ (rec(1 + 1) ∗ \underline{x}) = (\underline{x} + (−u(rec(1 + 1) ∗ \underline{x}))); EqNegated ⊳ (rec(1 + 1) ∗ \underline{x}) = (\underline{x} + (−u(rec(1 + 1) ∗ \underline{x}))) ≫ (−u(rec(1 + 1) ∗ \underline{x})) = (−u(\underline{x} + (−u(rec(1 + 1) ∗ \underline{x})))); MinusNegated ≫ (−u(\underline{x} + (−u(rec(1 + 1) ∗ \underline{x})))) = ((rec(1 + 1) ∗ \underline{x}) + (−u\underline{x})); eqTransitivity ⊳ (−u(rec(1 + 1) ∗ \underline{x})) = (−u(\underline{x} + (−u(rec(1+1)∗\underline{x}))))⊳(−u(\underline{x}+(−u(rec(1+1)∗\underline{x})))) = ((rec(1+1)∗\underline{x})+(−u\underline{x})) ≫ (−u(rec(1+1)∗\underline{x})) = ((rec(1+1)∗\underline{x})+(−u\underline{x})); eqSymmetry⊳(−u(rec(1+1)∗\underline{x})) = ((rec(1 + 1) ∗ \underline{x}) + (−u\underline{x})) ≫ ((rec(1 + 1) ∗ \underline{x}) + (−u\underline{x})) = (−u(rec(1 + 1) ∗ \underline{x})); EqAdditionLeft ⊳ ((rec(1 + 1) ∗ \underline{x}) + (−u\underline{x})) = (−u(rec(1 + 1) ∗ \underline{x})) ≫ ((rec(1 + 1) ∗ \underline{y}) + ((rec(1 + 1) ∗ \underline{x}) + (−u\underline{x}))) = ((rec(1 + 1) ∗ \underline{y}) + (−u(rec(1 + 1) ∗ \underline{x}))); DistributionOut(Minus) ≫ ((rec(1 + 1) ∗ \underline{y}) + (−u(rec(1 + 1) ∗ \underline{x}))) = (rec(1 + 1) ∗ (\underline{y} + (−u\underline{x}))); eqTransitivity6 ⊳ ((rec(1 + 1) ∗ (\underline{x} + \underline{y})) + (−u\underline{x})) = (((rec(1 + 1) ∗ \underline{x}) + (rec(1 + 1) ∗ \underline{y})) + (−u\underline{x})) ⊳ (((rec(1 + 1) ∗ \underline{x}) + (rec(1 + 1) ∗ \underline{y})) + (−u\underline{x})) = (((rec(1 + 1) ∗ \underline{y}) + (rec(1 + 1) ∗ \underline{x})) + (−u\underline{x})) ⊳ (((rec(1 + 1) ∗ \underline{y}) + (rec(1 + 1) ∗ \underline{x})) + (−u\underline{x})) = ((rec(1 + 1) ∗ \underline{y}) + ((rec(1 + 1) ∗ \underline{x}) + (−u\underline{x}))) ⊳ ((rec(1 + 1) ∗ \underline{y}) + ((rec(1 + 1) ∗ \underline{x}) + (−u\underline{x}))) = ((rec(1 + 1) ∗ \underline{y}) + (−u(rec(1 + 1) ∗ \underline{x}))) ⊳ ((rec(1 + 1) ∗ \underline{y}) + (−u(rec(1 + 1) ∗ \underline{x}))) = (rec(1 + 1) ∗ (\underline{y} + (−u\underline{x}))) ≫ ((rec(1 + 1) ∗ (\underline{x} + \underline{y})) + (−u\underline{x})) = (rec(1 + 1) ∗ (\underline{y} + (−u\underline{x})))⌉, p_{0}, c)]

[(1/2)(x + y) − x = (1/2)(y − x) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ((rec(1 + 1) ∗ (\underline{x} + \underline{y})) + (−u\underline{x})) = (rec(1 + 1) ∗ (\underline{y} + (−u\underline{x})))]

[(1/2)(x + y) − x = (1/2)(y − x) →^{tex} “(1/2)(x+y)-x=(1/2)(y-x)”]

[(1/2)(x + y) − x = (1/2)(y − x) →^{pyk} “lemma (1/2)(x+y)-x=(1/2)(y-x)”]

y − (1/2)(x + y) = (1/2)(y − x)

[y−(1/2)(x+y) = (1/2)(y−x) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: Distribution ≫ (rec(1 + 1) ∗ (\underline{x} + \underline{y})) = ((rec(1 + 1) ∗ \underline{x}) + (rec(1 + 1) ∗ \underline{y})); EqNegated ⊳ (rec(1 + 1) ∗ (\underline{x} + \underline{y})) = ((rec(1 + 1) ∗ \underline{x}) + (rec(1 + 1) ∗ \underline{y})) ≫ (−u(rec(1 + 1) ∗ (\underline{x} + \underline{y}))) = (−u((rec(1+1)∗\underline{x})+(rec(1+1)∗\underline{y}))); EqAdditionLeft⊳(−u(rec(1+1)∗(\underline{x}+\underline{y}))) = (−u((rec(1 + 1) ∗ \underline{x}) + (rec(1 + 1) ∗ \underline{y}))) ≫ (\underline{y} + (−u(rec(1 + 1) ∗ (\underline{x} + \underline{y})))) = (\underline{y} + (−u((rec(1 + 1) ∗ \underline{x}) + (rec(1 + 1) ∗ \underline{y})))); plusCommutativity ≫ ((rec(1 + 1) ∗ \underline{x}) + (rec(1 + 1) ∗ \underline{y})) = ((rec(1+1)∗\underline{y})+(rec(1+1)∗\underline{x})); EqNegated⊳((rec(1+1)∗\underline{x})+(rec(1+1)∗\underline{y})) = ((rec(1 + 1) ∗ \underline{y}) + (rec(1 + 1) ∗ \underline{x})) ≫ (−u((rec(1 + 1) ∗ \underline{x}) + (rec(1 + 1) ∗ \underline{y}))) = (−u((rec(1 + 1) ∗ \underline{y}) + (rec(1 + 1) ∗ \underline{x}))); −x − y = −(x + y) ≫ ((−u(rec(1 + 1) ∗ \underline{y})) + (−u(rec(1 + 1) ∗ \underline{x}))) = (−u((rec(1 + 1) ∗ \underline{y}) + (rec(1 + 1) ∗ \underline{x}))); eqSymmetry⊳((−u(rec(1+1)∗\underline{y}))+(−u(rec(1+1)∗\underline{x}))) = (−u((rec(1+1)∗ \underline{y})+(rec(1+1)∗\underline{x}))) ≫ (−u((rec(1+1)∗\underline{y})+(rec(1+1)∗\underline{x}))) = ((−u(rec(1+1)∗ \underline{y}))+(−u(rec(1+1)∗\underline{x}))); eqTransitivity⊳(−u((rec(1+1)∗\underline{x})+(rec(1+1)∗\underline{y}))) = (−u((rec(1 + 1) ∗ \underline{y}) + (rec(1 + 1) ∗ \underline{x}))) ⊳ (−u((rec(1 + 1) ∗ \underline{y}) + (rec(1 + 1) ∗ \underline{x}))) = ((−u(rec(1+1)∗\underline{y}))+(−u(rec(1+1)∗\underline{x}))) ≫ (−u((rec(1+1)∗\underline{x})+(rec(1+1)∗\underline{y}))) = ((−u(rec(1 + 1) ∗ \underline{y})) + (−u(rec(1 + 1) ∗ \underline{x}))); EqAdditionLeft ⊳ (−u((rec(1 + 1) ∗ \underline{x}) + (rec(1 + 1) ∗ \underline{y}))) = ((−u(rec(1 + 1) ∗ \underline{y})) + (−u(rec(1 + 1) ∗ \underline{x}))) ≫ (\underline{y}+(−u((rec(1+1)∗\underline{x})+(rec(1+1)∗\underline{y})))) = (\underline{y}+((−u(rec(1+1)∗\underline{y}))+(−u(rec(1+ 1)∗\underline{x})))); plusAssociativity ≫ ((\underline{y}+(−u(rec(1+1)∗\underline{y})))+(−u(rec(1+1)∗\underline{x}))) = (\underline{y}+((−u(rec(1+1)∗\underline{y}))+(−u(rec(1+1)∗\underline{x})))); eqSymmetry⊳((\underline{y}+(−u(rec(1+ 1)∗\underline{y})))+(−u(rec(1+1)∗\underline{x}))) = (\underline{y}+((−u(rec(1+1)∗\underline{y}))+(−u(rec(1+1)∗\underline{x})))) ≫ (\underline{y} + ((−u(rec(1 + 1) ∗ \underline{y})) + (−u(rec(1 + 1) ∗ \underline{x})))) = ((\underline{y} + (−u(rec(1 + 1) ∗ \underline{y}))) + (−u(rec(1 + 1) ∗ \underline{x}))); TwoHalves ≫ ((rec(1 + 1) ∗ \underline{y}) + (rec(1 + 1) ∗ \underline{y})) = \underline{y}; PositiveToRight(Eq)⊳((rec(1+1)∗\underline{y})+(rec(1+1)∗\underline{y})) = \underline{y} ≫ (rec(1+1)∗\underline{y}) = (\underline{y}+(−u(rec(1+1)∗\underline{y}))); eqSymmetry⊳(rec(1+1)∗\underline{y}) = (\underline{y}+(−u(rec(1+1)∗\underline{y}))) ≫ (\underline{y}+(−u(rec(1+1)∗\underline{y}))) = (rec(1+1)∗\underline{y}); eqAddition⊳(\underline{y}+(−u(rec(1+1)∗\underline{y}))) = (rec(1 + 1) ∗ \underline{y}) ≫ ((\underline{y} + (−u(rec(1 + 1) ∗ \underline{y}))) + (−u(rec(1 + 1) ∗ \underline{x}))) = ((rec(1 + 1) ∗ \underline{y}) + (−u(rec(1 + 1) ∗ \underline{x}))); DistributionOut(Minus) ≫ ((rec(1 + 1) ∗ \underline{y}) + (−u(rec(1 + 1) ∗ \underline{x}))) = (rec(1 + 1) ∗ (\underline{y} + (−u\underline{x}))); eqTransitivity6 ⊳ (\underline{y} + (−u(rec(1 + 1) ∗ (\underline{x} + \underline{y})))) = (\underline{y}+(−u((rec(1+1)∗\underline{x})+(rec(1+1)∗\underline{y}))))⊳(\underline{y}+(−u((rec(1+1)∗\underline{x})+(rec(1+1)∗ \underline{y})))) = (\underline{y} + ((−u(rec(1 + 1) ∗ \underline{y})) + (−u(rec(1 + 1) ∗ \underline{x})))) ⊳ (\underline{y} + ((−u(rec(1 + 1) ∗ \underline{y})) + (−u(rec(1 + 1) ∗ \underline{x})))) = ((\underline{y} + (−u(rec(1 + 1) ∗ \underline{y}))) + (−u(rec(1 + 1) ∗ \underline{x}))) ⊳ ((\underline{y} +(−u(rec(1+1)∗\underline{y})))+(−u(rec(1+1)∗\underline{x}))) = ((rec(1+1)∗\underline{y})+(−u(rec(1+ 1) ∗ \underline{x}))) ⊳ ((rec(1 + 1) ∗ \underline{y}) + (−u(rec(1 + 1) ∗ \underline{x}))) = (rec(1 + 1) ∗ (\underline{y} + (−u\underline{x}))) ≫ (\underline{y} + (−u(rec(1 + 1) ∗ (\underline{x} + \underline{y})))) = (rec(1 + 1) ∗ (\underline{y} + (−u\underline{x})))⌉, p_{0}, c)]

[y − (1/2)(x + y) = (1/2)(y − x) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: (\underline{y} + (−u(rec(1 + 1) ∗ (\underline{x} + \underline{y})))) = (rec(1 + 1) ∗ (\underline{y} + (−u\underline{x})))]

[y − (1/2)(x + y) = (1/2)(y − x) →^{tex} “y-(1/2)(x+y)=(1/2)(y-x)”]

[y − (1/2)(x + y) = (1/2)(y − x) →^{pyk} “lemma y-(1/2)(x+y)=(1/2)(y-x)”]

BSzero(Exact)

[BSzero(Exact) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: eqReflexivity ≫ 0 = 0; BSzero ⊳ 0 = 0 ≫ BS(\underline{m}, 0) = rec(1 + 1)(exp)\underline{m}⌉, p_{0}, c)]

[BSzero(Exact) →^{stmt} SystemQ ⊢ ∀\underline{m}: BS(\underline{m}, 0) = rec(1 + 1)(exp)\underline{m}]

[BSzero(Exact) →^{tex} “BSzero(Exact)”]

[BSzero(Exact) →^{pyk} “lemma base(1/2)Sum zero exact”]

SameBS(2)(Base)

[SameBS(2)(Base) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{(n2)}: 0 = \underline{(n2)} ⊢ eqSymmetry ⊳ 0 = \underline{(n2)} ≫ \underline{(n2)} = 0; BSzero ⊳ \underline{(n2)} = 0 ≫ BS(\underline{m}, \underline{(n2)}) = rec(1 + 1)(exp)\underline{m}; eqSymmetry ⊳ BS(\underline{m}, \underline{(n2)}) = rec(1 + 1)(exp)\underline{m} ≫ rec(1 + 1)(exp)\underline{m} = BS(\underline{m}, \underline{(n2)}); BSzero(Exact) ≫ BS(\underline{m}, 0) = rec(1 + 1)(exp)\underline{m}; eqTransitivity ⊳ BS(\underline{m}, 0) = rec(1 + 1)(exp)\underline{m} ⊳ rec(1 + 1)(exp)\underline{m} = BS(\underline{m}, \underline{(n2)}) ≫ BS(\underline{m}, 0) = BS(\underline{m}, \underline{(n2)}); ∀\underline{m}: ∀\underline{(n2)}: Ded ⊳ ∀\underline{m}: ∀\underline{(n2)}: 0 = \underline{(n2)} ⊢ BS(\underline{m}, 0) = BS(\underline{m}, \underline{(n2)}) ≫ 0 = \underline{(n2)} ⇒ BS(\underline{m}, 0) = BS(\underline{m}, \underline{(n2)}); Gen ⊳ 0 = \underline{(n2)} ⇒ BS(\underline{m}, 0) = BS(\underline{m}, \underline{(n2)}) ≫ ∀_{obj}\underline{(n2)}: 0 = \underline{(n2)} ⇒ BS(\underline{m}, 0) = BS(\underline{m}, \underline{(n2)})⌉, p_{0}, c)]

[SameBS(2)(Base) →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{(n2)}: ∀_{obj}\underline{(n2)}: 0 = \underline{(n2)} ⇒ BS(\underline{m}, 0) = BS(\underline{m}, \underline{(n2)})] [SameBS(2)(Base) →^{tex} “SameBS(2)(Base)”]

[SameBS(2)(Base) →^{pyk} “lemma sameBase(1/2)Sum second base”]

SameBS(2)(Indu)

[SameBS(2)(Indu) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{(n1)}: ∀\underline{(n2)}: ∀\underline{m}: ∀\underline{(n1)}: ∀\underline{(n2)}: ∀_{obj}\underline{(n2)}: \underline{(n1)} = \underline{(n2)} ⇒ BS(\underline{m}, \underline{(n1)}) = BS(\underline{m}, \underline{(n2)}) ⊢ (\underline{(n1)} + 1) = \underline{(n2)} ⊢ (+1)IsPositive(N) ≫ ˙¬ (0 <= (\underline{(n1)} + 1) ⇒ ˙¬ ( ˙¬ (0 = (\underline{(n1)} + 1))n)n)n; BSpositive ⊳ ˙¬ (0 <= (\underline{(n1)} + 1) ⇒ ˙¬ ( ˙¬ (0 = (\underline{(n1)} + 1))n)n)n ≫ BS(\underline{m}, (\underline{(n1)} + 1)) = (rec(1 + 1)(exp)(\underline{m} + (\underline{(n1)} + 1)) + BS(\underline{m}, ((\underline{(n1)} + 1) + (−u1)))); x = x + y − y ≫ \underline{(n1)} = ((\underline{(n1)} + 1) + (−u1)); A4 @((\underline{(n1)} + 1) + (−u1)) ⊳ ∀_{obj}\underline{(n2)}: \underline{(n1)} = \underline{(n2)} ⇒ BS(\underline{m}, \underline{(n1)}) = BS(\underline{m}, \underline{(n2)}) ≫ \underline{(n1)} = ((\underline{(n1)} + 1) + (−u1)) ⇒ BS(\underline{m}, \underline{(n1)}) = BS(\underline{m}, ((\underline{(n1)} + 1) + (−u1))); MP ⊳ \underline{(n1)} = ((\underline{(n1)} + 1) + (−u1)) ⇒ BS(\underline{m}, \underline{(n1)}) = BS(\underline{m}, ((\underline{(n1)} + 1) + (−u1))) ⊳ \underline{(n1)} = ((\underline{(n1)} + 1) + (−u1)) ≫ BS(\underline{m}, \underline{(n1)}) = BS(\underline{m}, ((\underline{(n1)} + 1) + (−u1))); eqSymmetry ⊳ BS(\underline{m}, \underline{(n1)}) = BS(\underline{m}, ((\underline{(n1)} + 1) + (−u1))) ≫ BS(\underline{m}, ((\underline{(n1)} + 1) + (−u1))) = BS(\underline{m}, \underline{(n1)}); EqAdditionLeft ⊳ BS(\underline{m}, ((\underline{(n1)} + 1) + (−u1))) = BS(\underline{m}, \underline{(n1)}) ≫ (rec(1 + 1)(exp)(\underline{m} + (\underline{(n1)} + 1)) + BS(\underline{m}, ((\underline{(n1)} + 1) + (−u1)))) = (rec(1+1)(exp)(\underline{m}+(\underline{(n1)}+1))+BS(\underline{m}, \underline{(n1)})); PositiveToRight(Eq)⊳(\underline{(n1)}+1) = \underline{(n2)} ≫ \underline{(n1)} = (\underline{(n2)} + (−u1)); A4 @(\underline{(n2)} + (−u1)) ⊳ ∀_{obj}\underline{(n2)}: \underline{(n1)} = \underline{(n2)} ⇒ BS(\underline{m}, \underline{(n1)}) = BS(\underline{m}, \underline{(n2)}) ≫ \underline{(n1)} = (\underline{(n2)} + (−u1)) ⇒ BS(\underline{m}, \underline{(n1)}) = BS(\underline{m}, (\underline{(n2)} + (−u1))); MP ⊳ \underline{(n1)} = (\underline{(n2)} + (−u1)) ⇒ BS(\underline{m}, \underline{(n1)}) = BS(\underline{m}, (\underline{(n2)} + (−u1))) ⊳ \underline{(n1)} = (\underline{(n2)} + (−u1)) ≫ BS(\underline{m}, \underline{(n1)}) = BS(\underline{m}, (\underline{(n2)} + (−u1))); EqAdditionLeft ⊳ BS(\underline{m}, \underline{(n1)}) = BS(\underline{m}, (\underline{(n2)} + (−u1))) ≫ (rec(1 + 1)(exp)(\underline{m} + (\underline{(n1)} + 1)) + BS(\underline{m}, \underline{(n1)})) = (rec(1 + 1)(exp)(\underline{m} + (\underline{(n1)} + 1)) + BS(\underline{m}, (\underline{(n2)} + (−u1)))); EqAdditionLeft ⊳ (\underline{(n1)}+1) = \underline{(n2)} ≫ (\underline{m}+(\underline{(n1)}+1)) = (\underline{m}+\underline{(n2)}); SameExp⊳(\underline{m}+(\underline{(n1)}+1)) = (\underline{m} + \underline{(n2)}) ≫ rec(1 + 1)(exp)(\underline{m} + (\underline{(n1)} + 1)) = rec(1 + 1)(exp)(\underline{m} + \underline{(n2)}); eqAddition⊳rec(1+1)(exp)(\underline{m}+(\underline{(n1)}+1)) = rec(1+1)(exp)(\underline{m}+\underline{(n2)}) ≫ (rec(1+1)(exp)(\underline{m}+(\underline{(n1)}+1))+BS(\underline{m}, (\underline{(n2)}+(−u1)))) = (rec(1+1)(exp)(\underline{m}+ \underline{(n2)}) + BS(\underline{m}, (\underline{(n2)} + (−u1)))); SubLessRight ⊳ (\underline{(n1)} + 1) = \underline{(n2)} ⊳ ˙¬ (0 <= (\underline{(n1)} + 1) ⇒ ˙¬ ( ˙¬ (0 = (\underline{(n1)} + 1))n)n)n ≫ ˙¬ (0 <= \underline{(n2)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(n2)})n)n)n; BSpositive⊳ ˙¬ (0 <= \underline{(n2)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(n2)})n)n)n ≫ BS(\underline{m}, \underline{(n2)}) = (rec(1+1)(exp)(\underline{m}+\underline{(n2)})+BS(\underline{m}, (\underline{(n2)}+(−u1)))); eqSymmetry⊳BS(\underline{m}, \underline{(n2)}) = (rec(1+1)(exp)(\underline{m}+\underline{(n2)})+BS(\underline{m}, (\underline{(n2)}+(−u1)))) ≫ (rec(1+1)(exp)(\underline{m}+\underline{(n2)})+ BS(\underline{m}, (\underline{(n2)} + (−u1)))) = BS(\underline{m}, \underline{(n2)}); eqTransitivity6 ⊳ BS(\underline{m}, (\underline{(n1)} + 1)) = (rec(1 + 1)(exp)(\underline{m} + (\underline{(n1)} + 1)) + BS(\underline{m}, ((\underline{(n1)} + 1) + (−u1)))) ⊳ (rec(1 + 1)(exp)(\underline{m} + (\underline{(n1)} + 1)) + BS(\underline{m}, ((\underline{(n1)} + 1) + (−u1)))) = (rec(1 + 1)(exp)(\underline{m} + (\underline{(n1)} + 1)) + BS(\underline{m}, \underline{(n1)})) ⊳ (rec(1 + 1)(exp)(\underline{m} + (\underline{(n1)} + 1)) + BS(\underline{m}, \underline{(n1)})) = (rec(1 + 1)(exp)(\underline{m} + (\underline{(n1)} + 1)) + BS(\underline{m}, (\underline{(n2)} + (−u1)))) ⊳ (rec(1 + 1)(exp)(\underline{m} + (\underline{(n1)}+1))+BS(\underline{m}, (\underline{(n2)}+(−u1)))) = (rec(1+1)(exp)(\underline{m}+\underline{(n2)})+BS(\underline{m}, (\underline{(n2)}+ (−u1))))⊳(rec(1+1)(exp)(\underline{m}+\underline{(n2)})+BS(\underline{m}, (\underline{(n2)}+(−u1)))) = BS(\underline{m}, \underline{(n2)}) ≫ BS(\underline{m}, (\underline{(n1)} + 1)) = BS(\underline{m}, \underline{(n2)}); Ded ⊳ ∀\underline{m}: ∀\underline{(n1)}: ∀\underline{(n2)}: ∀_{obj}\underline{(n2)}: \underline{(n1)} = \underline{(n2)} ⇒ BS(\underline{m}, \underline{(n1)}) = BS(\underline{m}, \underline{(n2)}) ⊢ (\underline{(n1)} + 1) = \underline{(n2)} ⊢ BS(\underline{m}, (\underline{(n1)} + 1)) = BS(\underline{m}, \underline{(n2)}) ≫ ∀_{obj}\underline{(n2)}: \underline{(n1)} = \underline{(n2)} ⇒ BS(\underline{m}, \underline{(n1)}) = BS(\underline{m}, \underline{(n2)}) ⇒ (\underline{(n1)} + 1) = \underline{(n2)} ⇒ BS(\underline{m}, (\underline{(n1)} + 1)) = BS(\underline{m}, \underline{(n2)}); ∀_{obj}\underline{(n2)}: \underline{(n1)} = \underline{(n2)} ⇒ BS(\underline{m}, \underline{(n1)}) = BS(\underline{m}, \underline{(n2)}) ⊢ MP ⊳ ∀_{obj}\underline{(n2)}: \underline{(n1)} = \underline{(n2)} ⇒ BS(\underline{m}, \underline{(n1)}) = BS(\underline{m}, \underline{(n2)}) ⇒ (\underline{(n1)} + 1) = \underline{(n2)} ⇒ BS(\underline{m}, (\underline{(n1)} + 1)) = BS(\underline{m}, \underline{(n2)}) ⊳ ∀_{obj}\underline{(n2)}: \underline{(n1)} = \underline{(n2)} ⇒ BS(\underline{m}, \underline{(n1)}) = BS(\underline{m}, \underline{(n2)}) ≫ (\underline{(n1)} + 1) = \underline{(n2)} ⇒ BS(\underline{m}, (\underline{(n1)} + 1)) = BS(\underline{m}, \underline{(n2)}); Gen ⊳ (\underline{(n1)} + 1) = \underline{(n2)} ⇒ BS(\underline{m}, (\underline{(n1)} + 1)) = BS(\underline{m}, \underline{(n2)}) ≫ ∀_{obj}\underline{(n2)}: (\underline{(n1)} + 1) = \underline{(n2)} ⇒ BS(\underline{m}, (\underline{(n1)} + 1)) = BS(\underline{m}, \underline{(n2)}); ∀\underline{m}: ∀\underline{(n1)}: ∀\underline{(n2)}: Ded ⊳ ∀\underline{m}: ∀\underline{(n1)}: ∀\underline{(n2)}: ∀_{obj}\underline{(n2)}: \underline{(n1)} = \underline{(n2)} ⇒ BS(\underline{m}, \underline{(n1)}) = BS(\underline{m}, \underline{(n2)}) ⊢ ∀_{obj}\underline{(n2)}: (\underline{(n1)} + 1) = \underline{(n2)} ⇒ BS(\underline{m}, (\underline{(n1)} + 1)) = BS(\underline{m}, \underline{(n2)}) ≫ ∀_{obj}\underline{(n2)}: \underline{(n1)} = \underline{(n2)} ⇒ BS(\underline{m}, \underline{(n1)}) = BS(\underline{m}, \underline{(n2)}) ⇒ ∀_{obj}\underline{(n2)}: (\underline{(n1)} + 1) = \underline{(n2)} ⇒ BS(\underline{m}, (\underline{(n1)} + 1)) = BS(\underline{m}, \underline{(n2)})⌉, p_{0}, c)]

[SameBS(2)(Indu) →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{(n1)}: ∀\underline{(n2)}: ∀_{obj}\underline{(n2)}: \underline{(n1)} = \underline{(n2)} ⇒ BS(\underline{m}, \underline{(n1)}) = BS(\underline{m}, \underline{(n2)}) ⇒ ∀_{obj}\underline{(n2)}: (\underline{(n1)} + 1) = \underline{(n2)} ⇒ BS(\underline{m}, (\underline{(n1)} + 1)) = BS(\underline{m}, \underline{(n2)})]

[SameBS(2)(Indu) →^{tex} “SameBS(2)(Indu)”]

[SameBS(2)(Indu) →^{pyk} “lemma sameBase(1/2)Sum second indu”]

SameBS(2)

[SameBS(2) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{(n1)}: ∀\underline{(n2)}: \underline{(n1)} = \underline{(n2)} ⊢ SameBS(2)(Base) ≫ ∀_{obj}\underline{(n2)}: 0 = \underline{(n2)} ⇒ BS(\underline{m}, 0) = BS(\underline{m}, \underline{(n2)}); SameBS(2)(Indu) ≫ ∀_{obj}\underline{(n2)}: \underline{(n1)} = \underline{(n2)} ⇒ BS(\underline{m}, \underline{(n1)}) = BS(\underline{m}, \underline{(n2)}) ⇒ ∀_{obj}\underline{(n2)}: (\underline{(n1)} + 1) = \underline{(n2)} ⇒ BS(\underline{m}, (\underline{(n1)} + 1)) = BS(\underline{m}, \underline{(n2)}); Induction ⊳ ∀_{obj}\underline{(n2)}: 0 = \underline{(n2)} ⇒ BS(\underline{m}, 0) = BS(\underline{m}, \underline{(n2)}) ⊳ ∀_{obj}\underline{(n2)}: \underline{(n1)} = \underline{(n2)} ⇒ BS(\underline{m}, \underline{(n1)}) = BS(\underline{m}, \underline{(n2)}) ⇒ ∀_{obj}\underline{(n2)}: (\underline{(n1)} + 1) = \underline{(n2)} ⇒ BS(\underline{m}, (\underline{(n1)} + 1)) = BS(\underline{m}, \underline{(n2)}) ≫ ∀_{obj}\underline{(n2)}: \underline{(n1)} = \underline{(n2)} ⇒ BS(\underline{m}, \underline{(n1)}) = BS(\underline{m}, \underline{(n2)}); A4 @ \underline{(n2)} ⊳ ∀_{obj}\underline{(n2)}: \underline{(n1)} = \underline{(n2)} ⇒ BS(\underline{m}, \underline{(n1)}) = BS(\underline{m}, \underline{(n2)}) ≫ \underline{(n1)} = \underline{(n2)} ⇒ BS(\underline{m}, \underline{(n1)}) = BS(\underline{m}, \underline{(n2)}); MP ⊳ \underline{(n1)} = \underline{(n2)} ⇒ BS(\underline{m}, \underline{(n1)}) = BS(\underline{m}, \underline{(n2)}) ⊳ \underline{(n1)} = \underline{(n2)} ≫ BS(\underline{m}, \underline{(n1)}) = BS(\underline{m}, \underline{(n2)})⌉, p_{0}, c)]

[SameBS(2) →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{(n1)}: ∀\underline{(n2)}: \underline{(n1)} = \underline{(n2)} ⊢ BS(\underline{m}, \underline{(n1)}) = BS(\underline{m}, \underline{(n2)})]

[SameBS(2) →^{tex} “SameBS(2)”]

[SameBS(2) →^{pyk} “lemma sameBase(1/2)Sum second”]

NegativeToLeft(Less)(1term)

[NegativeToLeft(Less)(1term) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ (0 <= (\underline{x} + (−u\underline{y})) ⇒ ˙¬ ( ˙¬ (0 = (\underline{x} + (−u\underline{y})))n)n)n ⊢ LessAddition ⊳ ˙¬ (0 <= (\underline{x} + (−u\underline{y})) ⇒ ˙¬ ( ˙¬ (0 = (\underline{x} + (−u\underline{y})))n)n)n ≫ ˙¬ ((0 + \underline{y}) <= ((\underline{x} + (−u\underline{y})) + \underline{y}) ⇒ ˙¬ ( ˙¬ ((0 + \underline{y}) = ((\underline{x} + (−u\underline{y})) + \underline{y}))n)n)n; plus0Left ≫ (0 + \underline{y}) = \underline{y}; SubLessLeft ⊳ (0 + \underline{y}) = \underline{y} ⊳ ˙¬ ((0 + \underline{y}) <= ((\underline{x} + (−u\underline{y})) + \underline{y}) ⇒ ˙¬ ( ˙¬ ((0 + \underline{y}) = ((\underline{x} + (−u\underline{y})) + \underline{y}))n)n)n ≫ ˙¬ (\underline{y} <= ((\underline{x} + (−u\underline{y})) + \underline{y}) ⇒ ˙¬ ( ˙¬ (\underline{y} = ((\underline{x} + (−u\underline{y})) + \underline{y}))n)n)n; Three2threeTerms ≫ ((\underline{x} + (−u\underline{y})) + \underline{y}) = ((\underline{x} + \underline{y}) + (−u\underline{y})); x = x + y − y ≫ \underline{x} = ((\underline{x} + \underline{y}) + (−u\underline{y})); eqSymmetry ⊳ \underline{x} = ((\underline{x} + \underline{y}) + (−u\underline{y})) ≫ ((\underline{x} + \underline{y}) + (−u\underline{y})) = \underline{x}; eqTransitivity ⊳ ((\underline{x} + (−u\underline{y})) + \underline{y}) = ((\underline{x} + \underline{y}) + (−u\underline{y})) ⊳ ((\underline{x} + \underline{y}) + (−u\underline{y})) = \underline{x} ≫ ((\underline{x} + (−u\underline{y})) + \underline{y}) = \underline{x}; SubLessRight ⊳ ((\underline{x} + (−u\underline{y})) + \underline{y}) = \underline{x} ⊳ ˙¬ (\underline{y} <= ((\underline{x} + (−u\underline{y})) + \underline{y}) ⇒ ˙¬ ( ˙¬ (\underline{y} = ((\underline{x} + (−u\underline{y})) + \underline{y}))n)n)n ≫ ˙¬ (\underline{y} <= \underline{x} ⇒ ˙¬ ( ˙¬ (\underline{y} = \underline{x})n)n)n⌉, p_{0}, c)]

[NegativeToLeft(Less)(1term) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ (0 <= (\underline{x} + (−u\underline{y})) ⇒ ˙¬ ( ˙¬ (0 = (\underline{x} + (−u\underline{y})))n)n)n ⊢ ˙¬ (\underline{y} <= \underline{x} ⇒ ˙¬ ( ˙¬ (\underline{y} = \underline{x})n)n)n]

[NegativeToLeft(Less)(1term) →^{tex} “NegativeToLeft(Less)(1 term)”]

[NegativeToLeft(Less)(1term) →^{pyk} “lemma negativeToLeft(Less)(1 term)”]

BS(+1)

[BS(+1) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: (+1)IsPositive(N) ≫ ˙¬ (0 <= (\underline{n} + 1) ⇒ ˙¬ ( ˙¬ (0 = (\underline{n} + 1))n)n)n; BSpositive ⊳ ˙¬ (0 <= (\underline{n} + 1) ⇒ ˙¬ ( ˙¬ (0 = (\underline{n} + 1))n)n)n ≫ BS(\underline{m}, (\underline{n} + 1)) = (rec(1 + 1)(exp)(\underline{m} + (\underline{n} + 1)) + BS(\underline{m}, ((\underline{n} + 1) + (−u1)))); plusAssociativity ≫ ((\underline{m} + \underline{n}) + 1) = (\underline{m} + (\underline{n} + 1)); SameExp ⊳ ((\underline{m} + \underline{n}) + 1) = (\underline{m} + (\underline{n} + 1)) ≫ rec(1 + 1)(exp)((\underline{m} + \underline{n}) + 1) = rec(1 + 1)(exp)(\underline{m} + (\underline{n} + 1)); eqSymmetry ⊳ rec(1 + 1)(exp)((\underline{m} + \underline{n}) + 1) = rec(1 + 1)(exp)(\underline{m} + (\underline{n} + 1)) ≫ rec(1 + 1)(exp)(\underline{m} + (\underline{n} + 1)) = rec(1 + 1)(exp)((\underline{m} + \underline{n}) + 1); x = x + y − y ≫ \underline{n} = ((\underline{n} + 1) + (−u1)); SameBS(2) ⊳ \underline{n} = ((\underline{n} + 1) + (−u1)) ≫ BS(\underline{m}, \underline{n}) = BS(\underline{m}, ((\underline{n} + 1) + (−u1))); eqSymmetry ⊳ BS(\underline{m}, \underline{n}) = BS(\underline{m}, ((\underline{n} + 1) + (−u1))) ≫ BS(\underline{m}, ((\underline{n} + 1) + (−u1))) = BS(\underline{m}, \underline{n}); AddEquations ⊳ rec(1 + 1)(exp)(\underline{m} + (\underline{n} + 1)) = rec(1 + 1)(exp)((\underline{m} + \underline{n}) + 1) ⊳ BS(\underline{m}, ((\underline{n} + 1) + (−u1))) = BS(\underline{m}, \underline{n}) ≫ (rec(1 + 1)(exp)(\underline{m} + (\underline{n} + 1)) + BS(\underline{m}, ((\underline{n} + 1) + (−u1)))) = (rec(1 + 1)(exp)((\underline{m} + \underline{n}) + 1) + BS(\underline{m}, \underline{n})); eqTransitivity ⊳ BS(\underline{m}, (\underline{n} + 1)) = (rec(1 + 1)(exp)(\underline{m} + (\underline{n} + 1)) + BS(\underline{m}, ((\underline{n} + 1) + (−u1)))) ⊳ (rec(1 + 1)(exp)(\underline{m} + (\underline{n}+1))+BS(\underline{m}, ((\underline{n}+1)+(−u1)))) = (rec(1+1)(exp)((\underline{m}+\underline{n})+1)+BS(\underline{m}, \underline{n})) ≫ BS(\underline{m}, (\underline{n} + 1)) = (rec(1 + 1)(exp)((\underline{m} + \underline{n}) + 1) + BS(\underline{m}, \underline{n}))⌉, p_{0}, c)]

[BS(+1) →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: BS(\underline{m}, (\underline{n} + 1)) = (rec(1 + 1)(exp)((\underline{m} + \underline{n}) + 1) + BS(\underline{m}, \underline{n}))]

[BS(+1) →^{tex} “BS(+1)”]

[BS(+1) →^{pyk} “lemma base(1/2)Sum(+1)”]

BSbound(Exact)(Base)

[BSbound(Exact)(Base) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: BSzero(Exact) ≫ BS((\underline{m} + 1), 0) = rec(1 + 1)(exp)(\underline{m} + 1); Exp(+1) ≫ rec(1 + 1)(exp)(\underline{m} + 1) = (rec(1 + 1) ∗ rec(1 + 1)(exp)\underline{m}); eqTransitivity ⊳ BS((\underline{m} + 1), 0) = rec(1+1)(exp)(\underline{m}+1)⊳rec(1+1)(exp)(\underline{m}+1) = (rec(1+1)∗rec(1+1)(exp)\underline{m}) ≫ BS((\underline{m} + 1), 0) = (rec(1 + 1) ∗ rec(1 + 1)(exp)\underline{m}); plus0 ≫ ((\underline{m} + 1) + 0) = (\underline{m} + 1); SameExp ⊳ ((\underline{m} + 1) + 0) = (\underline{m} + 1) ≫ rec(1 + 1)(exp)((\underline{m} + 1) + 0) = rec(1 + 1)(exp)(\underline{m} + 1); eqTransitivity ⊳ rec(1 + 1)(exp)((\underline{m} + 1) + 0) = rec(1+1)(exp)(\underline{m}+1)⊳rec(1+1)(exp)(\underline{m}+1) = (rec(1+1)∗rec(1+1)(exp)\underline{m}) ≫ rec(1 + 1)(exp)((\underline{m} + 1) + 0) = (rec(1 + 1) ∗ rec(1 + 1)(exp)\underline{m}); AddEquations ⊳ BS((\underline{m} + 1), 0) = (rec(1 + 1) ∗ rec(1 + 1)(exp)\underline{m}) ⊳ rec(1 + 1)(exp)((\underline{m} + 1) + 0) = (rec(1+1)∗rec(1+1)(exp)\underline{m}) ≫ (BS((\underline{m}+1), 0)+rec(1+1)(exp)((\underline{m}+1)+0)) = ((rec(1 + 1) ∗ rec(1 + 1)(exp)\underline{m}) + (rec(1 + 1) ∗ rec(1 + 1)(exp)\underline{m})); TwoHalves ≫ ((rec(1 + 1) ∗ rec(1 + 1)(exp)\underline{m}) + (rec(1 + 1) ∗ rec(1 + 1)(exp)\underline{m})) = rec(1+1)(exp)\underline{m}; eqTransitivity⊳(BS((\underline{m}+1), 0)+rec(1+1)(exp)((\underline{m}+1)+0)) = ((rec(1 + 1) ∗ rec(1 + 1)(exp)\underline{m}) + (rec(1 + 1) ∗ rec(1 + 1)(exp)\underline{m})) ⊳ ((rec(1 + 1) ∗ rec(1 + 1)(exp)\underline{m}) + (rec(1 + 1) ∗ rec(1 + 1)(exp)\underline{m})) = rec(1 + 1)(exp)\underline{m} ≫ (BS((\underline{m} + 1), 0) + rec(1 + 1)(exp)((\underline{m} + 1) + 0)) = rec(1 + 1)(exp)\underline{m}⌉, p_{0}, c)]

[BSbound(Exact)(Base) →^{stmt} SystemQ ⊢ ∀\underline{m}: (BS((\underline{m} + 1), 0) + rec(1 + 1)(exp)((\underline{m} + 1) + 0)) = rec(1 + 1)(exp)\underline{m}]

[BSbound(Exact)(Base) →^{tex} “BSbound(Exact)(Base)”]

[BSbound(Exact)(Base) →^{pyk} “lemma base(1/2)Sum exact bound base”]

BSbound(Exact)(Indu)

[BSbound(Exact)(Indu) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: (BS((\underline{m} + 1), \underline{n}) + rec(1 + 1)(exp)((\underline{m} + 1) + \underline{n})) = rec(1 + 1)(exp)\underline{m} ⊢ BS(+1) ≫ BS((\underline{m} + 1), (\underline{n} + 1)) = (rec(1 + 1)(exp)(((\underline{m} + 1) + \underline{n}) + 1) + BS((\underline{m} + 1), \underline{n})); eqAddition ⊳ BS((\underline{m} + 1), (\underline{n} + 1)) = (rec(1 + 1)(exp)(((\underline{m} + 1) + \underline{n}) + 1) + BS((\underline{m} + 1), \underline{n})) ≫ (BS((\underline{m} + 1), (\underline{n} + 1)) + rec(1 + 1)(exp)(((\underline{m} + 1) + \underline{n}) + 1)) = ((rec(1+1)(exp)(((\underline{m}+1)+\underline{n})+1)+BS((\underline{m}+1), \underline{n}))+rec(1+1)(exp)(((\underline{m}+1)+\underline{n})+ 1)); plusCommutativity ≫ (rec(1+1)(exp)(((\underline{m}+1)+\underline{n})+1)+BS((\underline{m}+1), \underline{n})) = (BS((\underline{m} + 1), \underline{n}) + rec(1 + 1)(exp)(((\underline{m} + 1) + \underline{n}) + 1)); eqAddition ⊳ (rec(1 + 1)(exp)(((\underline{m} + 1) + \underline{n}) + 1) + BS((\underline{m} + 1), \underline{n})) = (BS((\underline{m} + 1), \underline{n}) + rec(1 + 1)(exp)(((\underline{m} + 1) + \underline{n}) + 1)) ≫ ((rec(1 + 1)(exp)(((\underline{m} + 1) + \underline{n}) + 1) + BS((\underline{m} + 1), \underline{n})) + rec(1 + 1)(exp)(((\underline{m} + 1) + \underline{n}) + 1)) = ((BS((\underline{m}+1), \underline{n})+rec(1+1)(exp)(((\underline{m}+1)+\underline{n})+1))+rec(1+1)(exp)(((\underline{m}+1)+ \underline{n}) + 1)); Exp(+1) ≫ rec(1 + 1)(exp)(((\underline{m} + 1) + \underline{n}) + 1) = (rec(1 + 1) ∗ rec(1 + 1)(exp)((\underline{m} + 1) + \underline{n})); AddEquations ⊳ rec(1 + 1)(exp)(((\underline{m} + 1) + \underline{n}) + 1) = (rec(1 + 1) ∗ rec(1 + 1)(exp)((\underline{m} + 1) + \underline{n})) ⊳ rec(1 + 1)(exp)(((\underline{m} + 1) + \underline{n}) + 1) = (rec(1 + 1) ∗ rec(1 + 1)(exp)((\underline{m} + 1) + \underline{n})) ≫ (rec(1 + 1)(exp)(((\underline{m} + 1) + \underline{n}) + 1) + rec(1 + 1)(exp)(((\underline{m} + 1) + \underline{n}) + 1)) = ((rec(1 + 1) ∗ rec(1 + 1)(exp)((\underline{m} + 1) + \underline{n})) + (rec(1 + 1) ∗ rec(1 + 1)(exp)((\underline{m} + 1) + \underline{n}))); TwoHalves ≫ ((rec(1 + 1) ∗ rec(1 + 1)(exp)((\underline{m} + 1) + \underline{n})) + (rec(1 + 1) ∗ rec(1 + 1)(exp)((\underline{m} + 1) + \underline{n}))) = rec(1+1)(exp)((\underline{m}+1)+\underline{n}); eqTransitivity⊳(rec(1+1)(exp)(((\underline{m}+1)+\underline{n})+1)+ rec(1+1)(exp)(((\underline{m}+1)+\underline{n})+1)) = ((rec(1+1)∗rec(1+1)(exp)((\underline{m}+1)+\underline{n}))+ (rec(1+1)∗rec(1+1)(exp)((\underline{m}+1)+\underline{n})))⊳((rec(1+1)∗rec(1+1)(exp)((\underline{m}+1)+ \underline{n}))+(rec(1+1)∗rec(1+1)(exp)((\underline{m}+1)+\underline{n}))) = rec(1+1)(exp)((\underline{m}+1)+\underline{n}) ≫ (rec(1 + 1)(exp)(((\underline{m} + 1) + \underline{n}) + 1) + rec(1 + 1)(exp)(((\underline{m} + 1) + \underline{n}) + 1)) = rec(1 + 1)(exp)((\underline{m} + 1) + \underline{n}); Three2twoTerms ⊳ (rec(1 + 1)(exp)(((\underline{m} + 1) + \underline{n}) + 1)+rec(1+1)(exp)(((\underline{m}+1)+\underline{n})+1)) = rec(1+1)(exp)((\underline{m}+1)+\underline{n}) ≫ ((BS((\underline{m}+ 1), \underline{n}) + rec(1 + 1)(exp)(((\underline{m} + 1) + \underline{n}) + 1)) + rec(1 + 1)(exp)(((\underline{m} + 1) + \underline{n}) + 1)) = (BS((\underline{m} + 1), \underline{n}) + rec(1 + 1)(exp)((\underline{m} + 1) + \underline{n})); eqTransitivity5 ⊳ (BS((\underline{m} + 1), (\underline{n} + 1)) + rec(1 + 1)(exp)(((\underline{m} + 1) + \underline{n}) + 1)) = ((rec(1 + 1)(exp)(((\underline{m} + 1) + \underline{n}) + 1) + BS((\underline{m} + 1), \underline{n})) + rec(1 + 1)(exp)(((\underline{m} + 1) + \underline{n}) + 1)) ⊳ ((rec(1 + 1)(exp)(((\underline{m}+1)+\underline{n})+1)+BS((\underline{m}+1), \underline{n}))+rec(1+1)(exp)(((\underline{m}+1)+\underline{n})+1)) = ((BS((\underline{m} + 1), \underline{n}) + rec(1 + 1)(exp)(((\underline{m} + 1) + \underline{n}) + 1)) + rec(1 + 1)(exp)(((\underline{m} + 1) + \underline{n}) + 1)) ⊳ ((BS((\underline{m} + 1), \underline{n}) + rec(1 + 1)(exp)(((\underline{m} + 1) + \underline{n}) + 1)) + rec(1 + 1)(exp)(((\underline{m} + 1) + \underline{n}) + 1)) = (BS((\underline{m} + 1), \underline{n}) + rec(1 + 1)(exp)((\underline{m} + 1) + \underline{n})) ⊳ (BS((\underline{m} + 1), \underline{n}) + rec(1 + 1)(exp)((\underline{m} + 1) + \underline{n})) = rec(1 + 1)(exp)\underline{m} ≫ (BS((\underline{m} + 1), (\underline{n}+1))+rec(1+1)(exp)(((\underline{m}+1)+\underline{n})+1)) = rec(1+1)(exp)\underline{m}; ∀\underline{m}: ∀\underline{n}: Ded⊳ ∀\underline{m}: ∀\underline{n}: (BS((\underline{m} + 1), \underline{n}) + rec(1 + 1)(exp)((\underline{m} + 1) + \underline{n})) = rec(1 + 1)(exp)\underline{m} ⊢ (BS((\underline{m} + 1), (\underline{n} + 1)) + rec(1 + 1)(exp)(((\underline{m} + 1) + \underline{n}) + 1)) = rec(1 + 1)(exp)\underline{m} ≫ (BS((\underline{m} + 1), \underline{n}) + rec(1 + 1)(exp)((\underline{m} + 1) + \underline{n})) = rec(1 + 1)(exp)\underline{m} ⇒ (BS((\underline{m}+1), (\underline{n}+1))+rec(1+1)(exp)(((\underline{m}+1)+\underline{n})+1)) = rec(1+1)(exp)\underline{m}⌉, p_{0}, c)]

[BSbound(Exact)(Indu) →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: (BS((\underline{m} + 1), \underline{n}) + rec(1 + 1)(exp)((\underline{m} + 1) + \underline{n})) = rec(1 + 1)(exp)\underline{m} ⇒ (BS((\underline{m} + 1), (\underline{n} + 1)) + rec(1 + 1)(exp)(((\underline{m} + 1) + \underline{n}) + 1)) = rec(1 + 1)(exp)\underline{m}]

[BSbound(Exact)(Indu) →^{tex} “BSbound(Exact)(Indu)”]

[BSbound(Exact)(Indu) →^{pyk} “lemma base(1/2)Sum exact bound indu”]

BSbound(Exact)

[BSbound(Exact) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: BSbound(Exact)(Base) ≫ (BS((\underline{m} + 1), 0) + rec(1 + 1)(exp)((\underline{m} + 1) + 0)) = rec(1 + 1)(exp)\underline{m}; BSbound(Exact)(Indu) ≫ (BS((\underline{m} + 1), \underline{n}) + rec(1 + 1)(exp)((\underline{m} + 1) + \underline{n})) = rec(1 + 1)(exp)\underline{m} ⇒ (BS((\underline{m} + 1), (\underline{n} + 1)) + rec(1 + 1)(exp)(((\underline{m} + 1) + \underline{n}) + 1)) = rec(1 + 1)(exp)\underline{m}; Induction ⊳ (BS((\underline{m} + 1), 0) + rec(1 + 1)(exp)((\underline{m} + 1) + 0)) = rec(1 + 1)(exp)\underline{m} ⊳ (BS((\underline{m} + 1), \underline{n}) + rec(1 + 1)(exp)((\underline{m} + 1) + \underline{n})) = rec(1 + 1)(exp)\underline{m} ⇒ (BS((\underline{m} + 1), (\underline{n} + 1)) + rec(1 + 1)(exp)(((\underline{m} + 1) + \underline{n}) + 1)) = rec(1 + 1)(exp)\underline{m} ≫ (BS((\underline{m} + 1), \underline{n}) + rec(1 + 1)(exp)((\underline{m} + 1) + \underline{n})) = rec(1 + 1)(exp)\underline{m}⌉, p_{0}, c)]

[BSbound(Exact) →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: (BS((\underline{m} + 1), \underline{n}) + rec(1 + 1)(exp)((\underline{m} + 1) + \underline{n})) = rec(1 + 1)(exp)\underline{m}]

[BSbound(Exact) →^{tex} “BSbound(Exact)”]

[BSbound(Exact) →^{pyk} “lemma base(1/2)Sum exact bound”]

BSbound

[BSbound →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: BSbound(Exact) ≫ (BS((\underline{m} + 1), \underline{n}) + rec(1 + 1)(exp)((\underline{m} + 1) + \underline{n})) = rec(1 + 1)(exp)\underline{m}; plusCommutativity ≫ (rec(1 + 1)(exp)((\underline{m} + 1) + \underline{n}) + BS((\underline{m} + 1), \underline{n})) = (BS((\underline{m} + 1), \underline{n}) + rec(1 + 1)(exp)((\underline{m} + 1) + \underline{n})); eqTransitivity ⊳ (rec(1 + 1)(exp)((\underline{m} + 1) + \underline{n}) + BS((\underline{m} + 1), \underline{n})) = (BS((\underline{m} + 1), \underline{n}) + rec(1 + 1)(exp)((\underline{m} + 1) + \underline{n})) ⊳ (BS((\underline{m} + 1), \underline{n}) + rec(1 + 1)(exp)((\underline{m} + 1) + \underline{n})) = rec(1 + 1)(exp)\underline{m} ≫ (rec(1 + 1)(exp)((\underline{m} + 1) + \underline{n}) + BS((\underline{m} + 1), \underline{n})) = rec(1 + 1)(exp)\underline{m}; PositiveToRight(Eq) ⊳ (rec(1 + 1)(exp)((\underline{m} + 1) + \underline{n}) + BS((\underline{m} + 1), \underline{n})) = rec(1 + 1)(exp)\underline{m} ≫ rec(1 + 1)(exp)((\underline{m} + 1) + \underline{n}) = (rec(1 + 1)(exp)\underline{m} + (−uBS((\underline{m} + 1), \underline{n}))); 0 < 1/2 ≫ ˙¬ (0 <= rec(1 + 1) ⇒ ˙¬ ( ˙¬ (0 = rec(1 + 1))n)n)n; PositiveBase ⊳ ˙¬ (0 <= rec(1 + 1) ⇒ ˙¬ ( ˙¬ (0 = rec(1 + 1))n)n)n ≫ ˙¬ (0 <= rec(1 + 1)(exp)((\underline{m} + 1) + \underline{n}) ⇒ ˙¬ ( ˙¬ (0 = rec(1+1)(exp)((\underline{m}+1)+\underline{n}))n)n)n; SubLessRight⊳rec(1+1)(exp)((\underline{m}+1)+\underline{n}) = (rec(1 + 1)(exp)\underline{m} + (−uBS((\underline{m} + 1), \underline{n}))) ⊳ ˙¬ (0 <= rec(1 + 1)(exp)((\underline{m} + 1) + \underline{n}) ⇒ ˙¬ ( ˙¬ (0 = rec(1 + 1)(exp)((\underline{m} + 1) + \underline{n}))n)n)n ≫ ˙¬ (0 <= (rec(1 + 1)(exp)\underline{m} + (−uBS((\underline{m} + 1), \underline{n}))) ⇒ ˙¬ ( ˙¬ (0 = (rec(1 + 1)(exp)\underline{m} + (−uBS((\underline{m} + 1), \underline{n}))))n)n)n; NegativeToLeft(Less)(1term) ⊳ ˙¬ (0 <= (rec(1 + 1)(exp)\underline{m} + (−uBS((\underline{m} + 1), \underline{n}))) ⇒ ˙¬ ( ˙¬ (0 = (rec(1 + 1)(exp)\underline{m} + (−uBS((\underline{m} + 1), \underline{n}))))n)n)n ≫ ˙¬ (BS((\underline{m} + 1), \underline{n}) <= rec(1 + 1)(exp)\underline{m} ⇒ ˙¬ ( ˙¬ (BS((\underline{m} + 1), \underline{n}) = rec(1 + 1)(exp)\underline{m})n)n)n⌉, p_{0}, c)]

[BSbound →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: ˙¬ (BS((\underline{m} + 1), \underline{n}) <= rec(1 + 1)(exp)\underline{m} ⇒ ˙¬ ( ˙¬ (BS((\underline{m} + 1), \underline{n}) = rec(1 + 1)(exp)\underline{m})n)n)n]

[BSbound →^{tex} “BSbound”]

[BSbound →^{pyk} “lemma base(1/2)Sum bound”]

UStelescope(Zero)(Exact)

[UStelescope(Zero)(Exact) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: eqReflexivity ≫ 0 = 0; UStelescope(Zero) ⊳ 0 = 0 ≫ UStelescope(\underline{m}, 0) = |(us[\underline{m}] + (−uus[(\underline{m} + 1)]))|⌉, p_{0}, c)]

[UStelescope(Zero)(Exact) →^{stmt} SystemQ ⊢ ∀\underline{m}: UStelescope(\underline{m}, 0) = |(us[\underline{m}] + (−uus[(\underline{m} + 1)]))|]

[UStelescope(Zero)(Exact) →^{tex} “UStelescope(Zero)(Exact)”]

[UStelescope(Zero)(Exact) →^{pyk} “lemma UStelescope zero exact”]

SameTelescope(2)(Base)

[SameTelescope(2)(Base) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{(n2)}: 0 = \underline{(n2)} ⊢ eqSymmetry ⊳ 0 = \underline{(n2)} ≫ \underline{(n2)} = 0; UStelescope(Zero) ⊳ \underline{(n2)} = 0 ≫ UStelescope(\underline{m}, \underline{(n2)}) = |(us[\underline{m}] + (−uus[(\underline{m} + 1)]))|; eqSymmetry ⊳ UStelescope(\underline{m}, \underline{(n2)}) = |(us[\underline{m}] + (−uus[(\underline{m} + 1)]))| ≫ |(us[\underline{m}] + (−uus[(\underline{m} + 1)]))| = UStelescope(\underline{m}, \underline{(n2)}); UStelescope(Zero)(Exact) ≫ UStelescope(\underline{m}, 0) = |(us[\underline{m}] + (−uus[(\underline{m} + 1)]))|; eqTransitivity ⊳ UStelescope(\underline{m}, 0) = |(us[\underline{m}]+(−uus[(\underline{m}+1)]))|⊳|(us[\underline{m}]+(−uus[(\underline{m}+1)]))| = UStelescope(\underline{m}, \underline{(n2)}) ≫ UStelescope(\underline{m}, 0) = UStelescope(\underline{m}, \underline{(n2)}); ∀\underline{m}: ∀\underline{(n2)}: Ded ⊳ ∀\underline{m}: ∀\underline{(n2)}: 0 = \underline{(n2)} ⊢ UStelescope(\underline{m}, 0) = UStelescope(\underline{m}, \underline{(n2)}) ≫ 0 = \underline{(n2)} ⇒ UStelescope(\underline{m}, 0) = UStelescope(\underline{m}, \underline{(n2)}); Gen ⊳ 0 = \underline{(n2)} ⇒ UStelescope(\underline{m}, 0) = UStelescope(\underline{m}, \underline{(n2)}) ≫ ∀_{obj}\underline{(n2)}: 0 = \underline{(n2)} ⇒ UStelescope(\underline{m}, 0) = UStelescope(\underline{m}, \underline{(n2)})⌉, p_{0}, c)]

[SameTelescope(2)(Base) →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{(n2)}: ∀_{obj}\underline{(n2)}: 0 = \underline{(n2)} ⇒ UStelescope(\underline{m}, 0) = UStelescope(\underline{m}, \underline{(n2)})]

[SameTelescope(2)(Base) →^{tex} “SameTelescope(2)(Base)”]

[SameTelescope(2)(Base) →^{pyk} “lemma sameTelescope second base”]

SameTelescope(2)(Indu)

[SameTelescope(2)(Indu) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{(n1)}: ∀\underline{(n2)}: ∀\underline{m}: ∀\underline{(n1)}: ∀\underline{(n2)}: ∀_{obj}\underline{(n2)}: \underline{(n1)} = \underline{(n2)} ⇒ UStelescope(\underline{m}, \underline{(n1)}) = UStelescope(\underline{m}, \underline{(n2)}) ⊢ (\underline{(n1)} + 1) = \underline{(n2)} ⊢ (+1)IsPositive(N) ≫ ˙¬ (0 <= (\underline{(n1)} + 1) ⇒ ˙¬ ( ˙¬ (0 = (\underline{(n1)} + 1))n)n)n; UStelescope(Positive) ⊳ ˙¬ (0 <= (\underline{(n1)} + 1) ⇒ ˙¬ ( ˙¬ (0 = (\underline{(n1)} + 1))n)n)n ≫ UStelescope(\underline{m}, (\underline{(n1)} + 1)) = (|(us[(\underline{m} + (\underline{(n1)} + 1))] + (−uus[(\underline{m} + ((\underline{(n1)} + 1) + 1))]))| + UStelescope(\underline{m}, ((\underline{(n1)} + 1) + (−u1)))); x = x+y−y ≫ \underline{(n1)} = ((\underline{(n1)}+1)+(−u1)); A4 @((\underline{(n1)}+1)+(−u1))⊳∀_{obj}\underline{(n2)}: \underline{(n1)} = \underline{(n2)} ⇒ UStelescope(\underline{m}, \underline{(n1)}) = UStelescope(\underline{m}, \underline{(n2)}) ≫ \underline{(n1)} = ((\underline{(n1)} + 1) + (−u1)) ⇒ UStelescope(\underline{m}, \underline{(n1)}) = UStelescope(\underline{m}, ((\underline{(n1)} + 1) + (−u1))); MP ⊳ \underline{(n1)} = ((\underline{(n1)} + 1) + (−u1)) ⇒ UStelescope(\underline{m}, \underline{(n1)}) = UStelescope(\underline{m}, ((\underline{(n1)} + 1) + (−u1))) ⊳ \underline{(n1)} = ((\underline{(n1)} + 1) + (−u1)) ≫ UStelescope(\underline{m}, \underline{(n1)}) = UStelescope(\underline{m}, ((\underline{(n1)} + 1) + (−u1))); eqSymmetry ⊳ UStelescope(\underline{m}, \underline{(n1)}) = UStelescope(\underline{m}, ((\underline{(n1)} + 1) + (−u1))) ≫ UStelescope(\underline{m}, ((\underline{(n1)} + 1) + (−u1))) = UStelescope(\underline{m}, \underline{(n1)}); PositiveToRight(Eq) ⊳ (\underline{(n1)} + 1) = \underline{(n2)} ≫ \underline{(n1)} = (\underline{(n2)} + (−u1)); A4 @(\underline{(n2)} + (−u1)) ⊳ ∀_{obj}\underline{(n2)}: \underline{(n1)} = \underline{(n2)} ⇒ UStelescope(\underline{m}, \underline{(n1)}) = UStelescope(\underline{m}, \underline{(n2)}) ≫ \underline{(n1)} = (\underline{(n2)} + (−u1)) ⇒ UStelescope(\underline{m}, \underline{(n1)}) = UStelescope(\underline{m}, (\underline{(n2)} + (−u1))); MP ⊳ \underline{(n1)} = (\underline{(n2)} + (−u1)) ⇒ UStelescope(\underline{m}, \underline{(n1)}) = UStelescope(\underline{m}, (\underline{(n2)} + (−u1))) ⊳ \underline{(n1)} = (\underline{(n2)} + (−u1)) ≫ UStelescope(\underline{m}, \underline{(n1)}) = UStelescope(\underline{m}, (\underline{(n2)} + (−u1))); eqTransitivity ⊳ UStelescope(\underline{m}, ((\underline{(n1)} + 1) + (−u1))) = UStelescope(\underline{m}, \underline{(n1)}) ⊳ UStelescope(\underline{m}, \underline{(n1)}) = UStelescope(\underline{m}, (\underline{(n2)} + (−u1))) ≫ UStelescope(\underline{m}, ((\underline{(n1)} + 1) + (−u1))) = UStelescope(\underline{m}, (\underline{(n2)} + (−u1))); EqAdditionLeft ⊳ (\underline{(n1)} + 1) = \underline{(n2)} ≫ (\underline{m} + (\underline{(n1)} + 1)) = (\underline{m} + \underline{(n2)}); eqAddition ⊳ (\underline{(n1)} + 1) = \underline{(n2)} ≫ ((\underline{(n1)} + 1) + 1) = (\underline{(n2)} + 1); EqAdditionLeft ⊳ ((\underline{(n1)} + 1) + 1) = (\underline{(n2)} + 1) ≫ (\underline{m}+((\underline{(n1)}+1)+1)) = (\underline{m}+(\underline{(n2)}+1)); SameSeries(NumDiff)⊳(\underline{m}+(\underline{(n1)}+1)) = (\underline{m} + \underline{(n2)}) ⊳ (\underline{m} + ((\underline{(n1)} + 1) + 1)) = (\underline{m} + (\underline{(n2)} + 1)) ≫ |(us[(\underline{m} + (\underline{(n1)} + 1))] + (−uus[(\underline{m} + ((\underline{(n1)} + 1) + 1))]))| = |(us[(\underline{m}+\underline{(n2)})]+(−uus[(\underline{m}+(\underline{(n2)}+1))]))|; AddEquations⊳|(us[(\underline{m}+(\underline{(n1)}+1))]+ (−uus[(\underline{m} + ((\underline{(n1)} + 1) + 1))]))| = |(us[(\underline{m} + \underline{(n2)})] + (−uus[(\underline{m} + (\underline{(n2)} + 1))]))| ⊳ UStelescope(\underline{m}, ((\underline{(n1)} + 1) + (−u1))) = UStelescope(\underline{m}, (\underline{(n2)} + (−u1))) ≫ (|(us[(\underline{m} + (\underline{(n1)} + 1))] + (−uus[(\underline{m} + ((\underline{(n1)} + 1) + 1))]))| + UStelescope(\underline{m}, ((\underline{(n1)} + 1) + (−u1)))) = (|(us[(\underline{m} + \underline{(n2)})] + (−uus[(\underline{m} + (\underline{(n2)} + 1))]))| + UStelescope(\underline{m}, (\underline{(n2)} + (−u1)))); SubLessRight ⊳ (\underline{(n1)} + 1) = \underline{(n2)} ⊳ ˙¬ (0 <= (\underline{(n1)} + 1) ⇒ ˙¬ ( ˙¬ (0 = (\underline{(n1)} + 1))n)n)n ≫ ˙¬ (0 <= \underline{(n2)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(n2)})n)n)n; UStelescope(Positive) ⊳ ˙¬ (0 <= \underline{(n2)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(n2)})n)n)n ≫ UStelescope(\underline{m}, \underline{(n2)}) = (|(us[(\underline{m} + \underline{(n2)})] + (−uus[(\underline{m} + (\underline{(n2)} + 1))]))|+UStelescope(\underline{m}, (\underline{(n2)}+(−u1)))); eqSymmetry⊳UStelescope(\underline{m}, \underline{(n2)}) = (|(us[(\underline{m}+\underline{(n2)})]+(−uus[(\underline{m}+(\underline{(n2)}+1))]))|+UStelescope(\underline{m}, (\underline{(n2)}+(−u1)))) ≫ (|(us[(\underline{m}+\underline{(n2)})]+(−uus[(\underline{m}+(\underline{(n2)}+1))]))|+UStelescope(\underline{m}, (\underline{(n2)}+(−u1)))) = UStelescope(\underline{m}, \underline{(n2)}); eqTransitivity4 ⊳ UStelescope(\underline{m}, (\underline{(n1)} + 1)) = (|(us[(\underline{m} + (\underline{(n1)} + 1))] + (−uus[(\underline{m} + ((\underline{(n1)} + 1) + 1))]))| + UStelescope(\underline{m}, ((\underline{(n1)} + 1) + (−u1)))) ⊳ (|(us[(\underline{m} + (\underline{(n1)} + 1))] + (−uus[(\underline{m} + ((\underline{(n1)} + 1) + 1))]))| + UStelescope(\underline{m}, ((\underline{(n1)} + 1) + (−u1)))) = (|(us[(\underline{m}+\underline{(n2)})]+(−uus[(\underline{m}+(\underline{(n2)}+1))]))|+UStelescope(\underline{m}, (\underline{(n2)}+(−u1))))⊳ (|(us[(\underline{m}+\underline{(n2)})]+(−uus[(\underline{m}+(\underline{(n2)}+1))]))|+UStelescope(\underline{m}, (\underline{(n2)}+(−u1)))) = UStelescope(\underline{m}, \underline{(n2)}) ≫ UStelescope(\underline{m}, (\underline{(n1)} + 1)) = UStelescope(\underline{m}, \underline{(n2)}); Ded ⊳ ∀\underline{m}: ∀\underline{(n1)}: ∀\underline{(n2)}: ∀_{obj}\underline{(n2)}: \underline{(n1)} = \underline{(n2)} ⇒ UStelescope(\underline{m}, \underline{(n1)}) = UStelescope(\underline{m}, \underline{(n2)}) ⊢ (\underline{(n1)} + 1) = \underline{(n2)} ⊢ UStelescope(\underline{m}, (\underline{(n1)} + 1)) = UStelescope(\underline{m}, \underline{(n2)}) ≫ ∀_{obj}\underline{(n2)}: \underline{(n1)} = \underline{(n2)} ⇒ UStelescope(\underline{m}, \underline{(n1)}) = UStelescope(\underline{m}, \underline{(n2)}) ⇒ (\underline{(n1)} + 1) = \underline{(n2)} ⇒ UStelescope(\underline{m}, (\underline{(n1)} + 1)) = UStelescope(\underline{m}, \underline{(n2)}); ∀_{obj}\underline{(n2)}: \underline{(n1)} = \underline{(n2)} ⇒ UStelescope(\underline{m}, \underline{(n1)}) = UStelescope(\underline{m}, \underline{(n2)}) ⊢ MP ⊳ ∀_{obj}\underline{(n2)}: \underline{(n1)} = \underline{(n2)} ⇒ UStelescope(\underline{m}, \underline{(n1)}) = UStelescope(\underline{m}, \underline{(n2)}) ⇒ (\underline{(n1)} + 1) = \underline{(n2)} ⇒ UStelescope(\underline{m}, (\underline{(n1)} + 1)) = UStelescope(\underline{m}, \underline{(n2)}) ⊳ ∀_{obj}\underline{(n2)}: \underline{(n1)} = \underline{(n2)} ⇒ UStelescope(\underline{m}, \underline{(n1)}) = UStelescope(\underline{m}, \underline{(n2)}) ≫ (\underline{(n1)} + 1) = \underline{(n2)} ⇒ UStelescope(\underline{m}, (\underline{(n1)} + 1)) = UStelescope(\underline{m}, \underline{(n2)}); Gen ⊳ (\underline{(n1)} + 1) = \underline{(n2)} ⇒ UStelescope(\underline{m}, (\underline{(n1)} + 1)) = UStelescope(\underline{m}, \underline{(n2)}) ≫ ∀_{obj}\underline{(n2)}: (\underline{(n1)} + 1) = \underline{(n2)} ⇒ UStelescope(\underline{m}, (\underline{(n1)} + 1)) = UStelescope(\underline{m}, \underline{(n2)}); ∀\underline{m}: ∀\underline{(n1)}: ∀\underline{(n2)}: Ded ⊳ ∀\underline{m}: ∀\underline{(n1)}: ∀\underline{(n2)}: ∀_{obj}\underline{(n2)}: \underline{(n1)} = \underline{(n2)} ⇒ UStelescope(\underline{m}, \underline{(n1)}) = UStelescope(\underline{m}, \underline{(n2)}) ⊢ ∀_{obj}\underline{(n2)}: (\underline{(n1)} + 1) = \underline{(n2)} ⇒ UStelescope(\underline{m}, (\underline{(n1)} + 1)) = UStelescope(\underline{m}, \underline{(n2)}) ≫ ∀_{obj}\underline{(n2)}: \underline{(n1)} = \underline{(n2)} ⇒ UStelescope(\underline{m}, \underline{(n1)}) = UStelescope(\underline{m}, \underline{(n2)}) ⇒ ∀_{obj}\underline{(n2)}: (\underline{(n1)} + 1) = \underline{(n2)} ⇒ UStelescope(\underline{m}, (\underline{(n1)} + 1)) = UStelescope(\underline{m}, \underline{(n2)})⌉, p_{0}, c)]

[SameTelescope(2)(Indu) →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{(n1)}: ∀\underline{(n2)}: ∀_{obj}\underline{(n2)}: \underline{(n1)} = \underline{(n2)} ⇒ UStelescope(\underline{m}, \underline{(n1)}) = UStelescope(\underline{m}, \underline{(n2)}) ⇒ ∀_{obj}\underline{(n2)}: (\underline{(n1)} + 1) = \underline{(n2)} ⇒ UStelescope(\underline{m}, (\underline{(n1)} + 1)) = UStelescope(\underline{m}, \underline{(n2)})]

[SameTelescope(2)(Indu) →^{tex} “SameTelescope(2)(Indu)”]

[SameTelescope(2)(Indu) →^{pyk} “lemma sameTelescope second indu”]

SameTelescope(2)

[SameTelescope(2) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{(n1)}: ∀\underline{(n2)}: \underline{(n1)} = \underline{(n2)} ⊢ SameTelescope(2)(Base) ≫ ∀_{obj}\underline{(n2)}: 0 = \underline{(n2)} ⇒ UStelescope(\underline{m}, 0) = UStelescope(\underline{m}, \underline{(n2)}); SameTelescope(2)(Indu) ≫ ∀_{obj}\underline{(n2)}: \underline{(n1)} = \underline{(n2)} ⇒ UStelescope(\underline{m}, \underline{(n1)}) = UStelescope(\underline{m}, \underline{(n2)}) ⇒ ∀_{obj}\underline{(n2)}: (\underline{(n1)} + 1) = \underline{(n2)} ⇒ UStelescope(\underline{m}, (\underline{(n1)} + 1)) = UStelescope(\underline{m}, \underline{(n2)}); Induction ⊳ ∀_{obj}\underline{(n2)}: 0 = \underline{(n2)} ⇒ UStelescope(\underline{m}, 0) = UStelescope(\underline{m}, \underline{(n2)}) ⊳ ∀_{obj}\underline{(n2)}: \underline{(n1)} = \underline{(n2)} ⇒ UStelescope(\underline{m}, \underline{(n1)}) = UStelescope(\underline{m}, \underline{(n2)}) ⇒ ∀_{obj}\underline{(n2)}: (\underline{(n1)} + 1) = \underline{(n2)} ⇒ UStelescope(\underline{m}, (\underline{(n1)} + 1)) = UStelescope(\underline{m}, \underline{(n2)}) ≫ ∀_{obj}\underline{(n2)}: \underline{(n1)} = \underline{(n2)} ⇒ UStelescope(\underline{m}, \underline{(n1)}) = UStelescope(\underline{m}, \underline{(n2)}); A4 @ \underline{(n2)} ⊳ ∀_{obj}\underline{(n2)}: \underline{(n1)} = \underline{(n2)} ⇒ UStelescope(\underline{m}, \underline{(n1)}) = UStelescope(\underline{m}, \underline{(n2)}) ≫ \underline{(n1)} = \underline{(n2)} ⇒ UStelescope(\underline{m}, \underline{(n1)}) = UStelescope(\underline{m}, \underline{(n2)}); MP ⊳ \underline{(n1)} = \underline{(n2)} ⇒ UStelescope(\underline{m}, \underline{(n1)}) = UStelescope(\underline{m}, \underline{(n2)}) ⊳ \underline{(n1)} = \underline{(n2)} ≫ UStelescope(\underline{m}, \underline{(n1)}) = UStelescope(\underline{m}, \underline{(n2)})⌉, p_{0}, c)]

[SameTelescope(2) →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{(n1)}: ∀\underline{(n2)}: \underline{(n1)} = \underline{(n2)} ⊢ UStelescope(\underline{m}, \underline{(n1)}) = UStelescope(\underline{m}, \underline{(n2)})]

[SameTelescope(2) →^{tex} “SameTelescope(2)”]

[SameTelescope(2) →^{pyk} “lemma sameTelescope second”]

Exp(+1)

[Exp(+1) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{x}: (+1)IsPositive(N) ≫ ˙¬ (0 <= (\underline{m} + 1) ⇒ ˙¬ ( ˙¬ (0 = (\underline{m} + 1))n)n)n; ExpPositive ⊳ ˙¬ (0 <= (\underline{m} + 1) ⇒ ˙¬ ( ˙¬ (0 = (\underline{m} + 1))n)n)n ≫ \underline{x}(exp)(\underline{m} + 1) = (\underline{x} ∗ \underline{x}(exp)((\underline{m} + 1) + (−u1))); x = x + y − y ≫ \underline{m} = ((\underline{m} + 1) + (−u1)); SameExp ⊳ \underline{m} = ((\underline{m} + 1) + (−u1)) ≫ \underline{x}(exp)\underline{m} = \underline{x}(exp)((\underline{m} + 1) + (−u1)); EqMultiplicationLeft ⊳ \underline{x}(exp)\underline{m} = \underline{x}(exp)((\underline{m} + 1) + (−u1)) ≫ (\underline{x} ∗ \underline{x}(exp)\underline{m}) = (\underline{x} ∗ \underline{x}(exp)((\underline{m} + 1) + (−u1))); eqSymmetry ⊳ (\underline{x} ∗ \underline{x}(exp)\underline{m}) = (\underline{x} ∗ \underline{x}(exp)((\underline{m} + 1) + (−u1))) ≫ (\underline{x} ∗ \underline{x}(exp)((\underline{m} + 1) + (−u1))) = (\underline{x} ∗ \underline{x}(exp)\underline{m}); eqTransitivity ⊳ \underline{x}(exp)(\underline{m} + 1) = (\underline{x} ∗ \underline{x}(exp)((\underline{m} + 1) + (−u1))) ⊳ (\underline{x} ∗ \underline{x}(exp)((\underline{m} + 1) + (−u1))) = (\underline{x} ∗ \underline{x}(exp)\underline{m}) ≫ \underline{x}(exp)(\underline{m} + 1) = (\underline{x} ∗ \underline{x}(exp)\underline{m})⌉, p_{0}, c)]

[Exp(+1) →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{x}: \underline{x}(exp)(\underline{m} + 1) = (\underline{x} ∗ \underline{x}(exp)\underline{m})]

[Exp(+1) →^{tex} “Exp(+1)”]

[Exp(+1) →^{pyk} “lemma exp(+1)”]

PositiveBase(Base)

[PositiveBase(Base) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ExpZero(Exact) ≫ \underline{x}(exp)0 = 1; eqSymmetry ⊳ \underline{x}(exp)0 = 1 ≫ 1 = \underline{x}(exp)0; 0 < 1 ≫ ˙¬ (0 <= 1 ⇒ ˙¬ ( ˙¬ (0 = 1)n)n)n; SubLessRight ⊳ 1 = \underline{x}(exp)0 ⊳ ˙¬ (0 <= 1 ⇒ ˙¬ ( ˙¬ (0 = 1)n)n)n ≫ ˙¬ (0 <= \underline{x}(exp)0 ⇒ ˙¬ ( ˙¬ (0 = \underline{x}(exp)0)n)n)n⌉, p_{0}, c)]

[PositiveBase(Base) →^{stmt} SystemQ ⊢ ∀\underline{x}: ˙¬ (0 <= \underline{x}(exp)0 ⇒ ˙¬ ( ˙¬ (0 = \underline{x}(exp)0)n)n)n]

[PositiveBase(Base) →^{tex} “PositiveBase(Base)”]

[PositiveBase(Base) →^{pyk} “lemma positiveBase base”]

PositiveBase(Indu)

[PositiveBase(Indu) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{x}: ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ⊢ ˙¬ (0 <= \underline{x}(exp)\underline{m} ⇒ ˙¬ ( ˙¬ (0 = \underline{x}(exp)\underline{m})n)n)n ⊢ Exp(+1) ≫ \underline{x}(exp)(\underline{m} + 1) = (\underline{x} ∗ \underline{x}(exp)\underline{m}); eqSymmetry ⊳ \underline{x}(exp)(\underline{m} + 1) = (\underline{x} ∗ \underline{x}(exp)\underline{m}) ≫ (\underline{x} ∗ \underline{x}(exp)\underline{m}) = \underline{x}(exp)(\underline{m} + 1); PositiveFactors ⊳ ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ⊳ ˙¬ (0 <= \underline{x}(exp)\underline{m} ⇒ ˙¬ ( ˙¬ (0 = \underline{x}(exp)\underline{m})n)n)n ≫ ˙¬ (0 <= (\underline{x} ∗ \underline{x}(exp)\underline{m}) ⇒ ˙¬ ( ˙¬ (0 = (\underline{x} ∗ \underline{x}(exp)\underline{m}))n)n)n; SubLessRight ⊳ (\underline{x} ∗ \underline{x}(exp)\underline{m}) = \underline{x}(exp)(\underline{m} + 1) ⊳ ˙¬ (0 <= (\underline{x} ∗ \underline{x}(exp)\underline{m}) ⇒ ˙¬ ( ˙¬ (0 = (\underline{x} ∗ \underline{x}(exp)\underline{m}))n)n)n ≫ ˙¬ (0 <= \underline{x}(exp)(\underline{m} + 1) ⇒ ˙¬ ( ˙¬ (0 = \underline{x}(exp)(\underline{m} + 1))n)n)n; ∀\underline{m}: ∀\underline{x}: Ded ⊳ ∀\underline{m}: ∀\underline{x}: ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ⊢ ˙¬ (0 <= \underline{x}(exp)\underline{m} ⇒ ˙¬ ( ˙¬ (0 = \underline{x}(exp)\underline{m})n)n)n ⊢ ˙¬ (0 <= \underline{x}(exp)(\underline{m} + 1) ⇒ ˙¬ ( ˙¬ (0 = \underline{x}(exp)(\underline{m} + 1))n)n)n ≫ ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ⇒ ˙¬ (0 <= \underline{x}(exp)\underline{m} ⇒ ˙¬ ( ˙¬ (0 = \underline{x}(exp)\underline{m})n)n)n ⇒ ˙¬ (0 <= \underline{x}(exp)(\underline{m} + 1) ⇒ ˙¬ ( ˙¬ (0 = \underline{x}(exp)(\underline{m} + 1))n)n)n; ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ⊢ MP ⊳ ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ⇒ ˙¬ (0 <= \underline{x}(exp)\underline{m} ⇒ ˙¬ ( ˙¬ (0 = \underline{x}(exp)\underline{m})n)n)n ⇒ ˙¬ (0 <= \underline{x}(exp)(\underline{m} + 1) ⇒ ˙¬ ( ˙¬ (0 = \underline{x}(exp)(\underline{m} + 1))n)n)n ⊳ ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ≫ ˙¬ (0 <= \underline{x}(exp)\underline{m} ⇒ ˙¬ ( ˙¬ (0 = \underline{x}(exp)\underline{m})n)n)n ⇒ ˙¬ (0 <= \underline{x}(exp)(\underline{m} + 1) ⇒ ˙¬ ( ˙¬ (0 = \underline{x}(exp)(\underline{m} + 1))n)n)n⌉, p_{0}, c)]

[PositiveBase(Indu) →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{x}: ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ⊢ ˙¬ (0 <= \underline{x}(exp)\underline{m} ⇒ ˙¬ ( ˙¬ (0 = \underline{x}(exp)\underline{m})n)n)n ⇒ ˙¬ (0 <= \underline{x}(exp)(\underline{m} + 1) ⇒ ˙¬ ( ˙¬ (0 = \underline{x}(exp)(\underline{m} + 1))n)n)n]

[PositiveBase(Indu) →^{tex} “PositiveBase(Indu)”]

[PositiveBase(Indu) →^{pyk} “lemma positiveBase indu”]

PositiveBase

[PositiveBase →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{x}: ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ⊢ PositiveBase(Base) ≫ ˙¬ (0 <= \underline{x}(exp)0 ⇒ ˙¬ ( ˙¬ (0 = \underline{x}(exp)0)n)n)n; PositiveBase(Indu) ⊳ ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ≫ ˙¬ (0 <= \underline{x}(exp)\underline{m} ⇒ ˙¬ ( ˙¬ (0 = \underline{x}(exp)\underline{m})n)n)n ⇒ ˙¬ (0 <= \underline{x}(exp)(\underline{m} + 1) ⇒ ˙¬ ( ˙¬ (0 = \underline{x}(exp)(\underline{m} + 1))n)n)n; Induction ⊳ ˙¬ (0 <= \underline{x}(exp)0 ⇒ ˙¬ ( ˙¬ (0 = \underline{x}(exp)0)n)n)n ⊳ ˙¬ (0 <= \underline{x}(exp)\underline{m} ⇒ ˙¬ ( ˙¬ (0 = \underline{x}(exp)\underline{m})n)n)n ⇒ ˙¬ (0 <= \underline{x}(exp)(\underline{m} + 1) ⇒ ˙¬ ( ˙¬ (0 = \underline{x}(exp)(\underline{m} + 1))n)n)n ≫ ˙¬ (0 <= \underline{x}(exp)\underline{m} ⇒ ˙¬ ( ˙¬ (0 = \underline{x}(exp)\underline{m})n)n)n⌉, p_{0}, c)]

[PositiveBase →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{x}: ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ⊢ ˙¬ (0 <= \underline{x}(exp)\underline{m} ⇒ ˙¬ ( ˙¬ (0 = \underline{x}(exp)\underline{m})n)n)n]

[PositiveBase →^{tex} “PositiveBase”]

[PositiveBase →^{pyk} “lemma positiveBase”]

TelescopeNumerical(Base)

[TelescopeNumerical(Base) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: eqReflexivity ≫ 0 = 0; UStelescope(Zero) ⊳ 0 = 0 ≫ UStelescope(\underline{m}, 0) = |(us[\underline{m}] + (−uus[(\underline{m} + 1)]))|; eqReflexivity ≫ \underline{m} = \underline{m}; plus0Left ≫ (0 + 1) = 1; EqAdditionLeft ⊳ (0 + 1) = 1 ≫ (\underline{m} + (0 + 1)) = (\underline{m} + 1); eqSymmetry ⊳ (\underline{m} + (0 + 1)) = (\underline{m} + 1) ≫ (\underline{m} + 1) = (\underline{m} + (0 + 1)); SameSeries(NumDiff) ⊳ \underline{m} = \underline{m} ⊳ (\underline{m} + 1) = (\underline{m} + (0 + 1)) ≫ |(us[\underline{m}] + (−uus[(\underline{m} + 1)]))| = |(us[\underline{m}] + (−uus[(\underline{m} + (0 + 1))]))|; eqTransitivity ⊳ UStelescope(\underline{m}, 0) = |(us[\underline{m}] + (−uus[(\underline{m} + 1)]))| ⊳ |(us[\underline{m}] + (−uus[(\underline{m} + 1)]))| = |(us[\underline{m}] + (−uus[(\underline{m} + (0 + 1))]))| ≫ UStelescope(\underline{m}, 0) = |(us[\underline{m}] + (−uus[(\underline{m} + (0 + 1))]))|; eqSymmetry ⊳ UStelescope(\underline{m}, 0) = |(us[\underline{m}] + (−uus[(\underline{m} + (0 + 1))]))| ≫ |(us[\underline{m}] + (−uus[(\underline{m} + (0 + 1))]))| = UStelescope(\underline{m}, 0); eqLeq ⊳ |(us[\underline{m}] + (−uus[(\underline{m} + (0 + 1))]))| = UStelescope(\underline{m}, 0) ≫ |(us[\underline{m}] + (−uus[(\underline{m} + (0 + 1))]))| <= UStelescope(\underline{m}, 0)⌉, p_{0}, c)]

[TelescopeNumerical(Base) →^{stmt} SystemQ ⊢ ∀\underline{m}: |(us[\underline{m}] + (−uus[(\underline{m} + (0 + 1))]))| <= UStelescope(\underline{m}, 0)]

[TelescopeNumerical(Base) →^{tex} “TelescopeNumerical(Base)”]

[TelescopeNumerical(Base) →^{pyk} “lemma telescopeNumerical base”]

TelescopeNumerical(Indu)

[TelescopeNumerical(Indu) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: |(us[\underline{m}] + (−uus[(\underline{m} + (\underline{n} + 1))]))| <= UStelescope(\underline{m}, \underline{n}) ⊢ (+1)IsPositive(N) ≫ ˙¬ (0 <= (\underline{n} + 1) ⇒ ˙¬ ( ˙¬ (0 = (\underline{n} + 1))n)n)n; UStelescope(Positive) ⊳ ˙¬ (0 <= (\underline{n} + 1) ⇒ ˙¬ ( ˙¬ (0 = (\underline{n} + 1))n)n)n ≫ UStelescope(\underline{m}, (\underline{n} + 1)) = (|(us[(\underline{m} + (\underline{n} + 1))] + (−uus[(\underline{m} + ((\underline{n} + 1) + 1))]))| + UStelescope(\underline{m}, ((\underline{n} + 1) + (−u1)))); x = x + y − y ≫ \underline{n} = ((\underline{n} + 1) + (−u1)); eqSymmetry ⊳ \underline{n} = ((\underline{n} + 1) + (−u1)) ≫ ((\underline{n} + 1) + (−u1)) = \underline{n}; SameTelescope(2) ⊳ ((\underline{n} + 1) + (−u1)) = \underline{n} ≫ UStelescope(\underline{m}, ((\underline{n} + 1) + (−u1))) = UStelescope(\underline{m}, \underline{n}); EqAdditionLeft ⊳ UStelescope(\underline{m}, ((\underline{n} + 1) + (−u1))) = UStelescope(\underline{m}, \underline{n}) ≫ (|(us[(\underline{m} + (\underline{n} + 1))] + (−uus[(\underline{m} + ((\underline{n} + 1) + 1))]))| + UStelescope(\underline{m}, ((\underline{n} + 1) + (−u1)))) = (|(us[(\underline{m} + (\underline{n} + 1))] + (−uus[(\underline{m} + ((\underline{n} + 1) + 1))]))| + UStelescope(\underline{m}, \underline{n})); eqTransitivity ⊳ UStelescope(\underline{m}, (\underline{n} + 1)) = (|(us[(\underline{m} + (\underline{n} + 1))]+(−uus[(\underline{m}+((\underline{n}+1)+1))]))|+UStelescope(\underline{m}, ((\underline{n}+1)+(−u1))))⊳(|(us[(\underline{m}+ (\underline{n} + 1))] + (−uus[(\underline{m} + ((\underline{n} + 1) + 1))]))| + UStelescope(\underline{m}, ((\underline{n} + 1) + (−u1)))) = (|(us[(\underline{m} + (\underline{n} + 1))] + (−uus[(\underline{m} + ((\underline{n} + 1) + 1))]))| + UStelescope(\underline{m}, \underline{n})) ≫ UStelescope(\underline{m}, (\underline{n} + 1)) = (|(us[(\underline{m} + (\underline{n} + 1))] + (−uus[(\underline{m} + ((\underline{n} + 1) + 1))]))| + UStelescope(\underline{m}, \underline{n})); eqSymmetry ⊳ UStelescope(\underline{m}, (\underline{n} + 1)) = (|(us[(\underline{m} + (\underline{n} + 1))] + (−uus[(\underline{m} + ((\underline{n} + 1) + 1))]))| + UStelescope(\underline{m}, \underline{n})) ≫ (|(us[(\underline{m} + (\underline{n} + 1))] + (−uus[(\underline{m} + ((\underline{n} + 1) + 1))]))| + UStelescope(\underline{m}, \underline{n})) = UStelescope(\underline{m}, (\underline{n} + 1)); LeqAdditionLeft ⊳ |(us[\underline{m}] + (−uus[(\underline{m} + (\underline{n} + 1))]))| <= UStelescope(\underline{m}, \underline{n}) ≫ (|(us[(\underline{m} + (\underline{n} + 1))] + (−uus[(\underline{m} + ((\underline{n} + 1) + 1))]))| + |(us[\underline{m}]+(−uus[(\underline{m}+(\underline{n}+1))]))|) <= (|(us[(\underline{m}+(\underline{n}+1))]+(−uus[(\underline{m}+((\underline{n}+1)+ 1))]))| + UStelescope(\underline{m}, \underline{n})); subLeqRight ⊳ (|(us[(\underline{m} + (\underline{n} + 1))] + (−uus[(\underline{m} + ((\underline{n} + 1) + 1))]))| + UStelescope(\underline{m}, \underline{n})) = UStelescope(\underline{m}, (\underline{n} + 1)) ⊳ (|(us[(\underline{m} + (\underline{n} + 1))] + (−uus[(\underline{m} + ((\underline{n} + 1) + 1))]))| + |(us[\underline{m}] + (−uus[(\underline{m} + (\underline{n} + 1))]))|) <= (|(us[(\underline{m} + (\underline{n} + 1))] + (−uus[(\underline{m} + ((\underline{n} + 1) + 1))]))| + UStelescope(\underline{m}, \underline{n})) ≫ (|(us[(\underline{m} + (\underline{n} + 1))] + (−uus[(\underline{m} + ((\underline{n} + 1) + 1))]))| + |(us[\underline{m}] + (−uus[(\underline{m} + (\underline{n} + 1))]))|) <= UStelescope(\underline{m}, (\underline{n} + 1)); plusCommutativity ≫ (|(us[(\underline{m} + (\underline{n} + 1))] + (−uus[(\underline{m} + ((\underline{n} + 1) + 1))]))| + |(us[\underline{m}] + (−uus[(\underline{m} + (\underline{n} + 1))]))|) = (|(us[\underline{m}] + (−uus[(\underline{m} + (\underline{n} + 1))]))| + |(us[(\underline{m} + (\underline{n} + 1))] + (−uus[(\underline{m} + ((\underline{n} + 1) + 1))]))|); subLeqLeft ⊳ (|(us[(\underline{m} + (\underline{n} + 1))] + (−uus[(\underline{m} + ((\underline{n} + 1) + 1))]))| + |(us[\underline{m}]+(−uus[(\underline{m}+(\underline{n}+1))]))|) = (|(us[\underline{m}]+(−uus[(\underline{m}+(\underline{n}+1))]))|+|(us[(\underline{m}+ (\underline{n} + 1))] + (−uus[(\underline{m} + ((\underline{n} + 1) + 1))]))|) ⊳ (|(us[(\underline{m} + (\underline{n} + 1))] + (−uus[(\underline{m} + ((\underline{n} + 1) + 1))]))| + |(us[\underline{m}] + (−uus[(\underline{m} + (\underline{n} + 1))]))|) <= UStelescope(\underline{m}, (\underline{n} + 1)) ≫ (|(us[\underline{m}] + (−uus[(\underline{m} + (\underline{n} + 1))]))| + |(us[(\underline{m} + (\underline{n} + 1))] + (−uus[(\underline{m} + ((\underline{n} + 1) + 1))]))|) <= UStelescope(\underline{m}, (\underline{n} + 1)); insertMiddleTerm(Numerical) ≫ |(us[\underline{m}] + (−uus[(\underline{m} + ((\underline{n} + 1) + 1))]))| <= (|(us[\underline{m}] + (−uus[(\underline{m} + (\underline{n} + 1))]))| + |(us[(\underline{m} + (\underline{n} + 1))] + (−uus[(\underline{m} + ((\underline{n} + 1) + 1))]))|); leqTransitivity ⊳ |(us[\underline{m}] + (−uus[(\underline{m} + ((\underline{n} + 1) + 1))]))| <= (|(us[\underline{m}] + (−uus[(\underline{m}+(\underline{n}+1))]))|+|(us[(\underline{m}+(\underline{n}+1))]+(−uus[(\underline{m}+((\underline{n}+1)+1))]))|)⊳(|(us[\underline{m}]+ (−uus[(\underline{m} + (\underline{n} + 1))]))| + |(us[(\underline{m} + (\underline{n} + 1))] + (−uus[(\underline{m} + ((\underline{n} + 1) + 1))]))|) <= UStelescope(\underline{m}, (\underline{n} + 1)) ≫ |(us[\underline{m}] + (−uus[(\underline{m} + ((\underline{n} + 1) + 1))]))| <= UStelescope(\underline{m}, (\underline{n}+1)); ∀\underline{m}: ∀\underline{n}: Ded⊳∀\underline{m}: ∀\underline{n}: |(us[\underline{m}]+(−uus[(\underline{m}+(\underline{n}+1))]))| <= UStelescope(\underline{m}, \underline{n}) ⊢ |(us[\underline{m}] + (−uus[(\underline{m} + ((\underline{n} + 1) + 1))]))| <= UStelescope(\underline{m}, (\underline{n} + 1)) ≫ |(us[\underline{m}] + (−uus[(\underline{m} + (\underline{n} + 1))]))| <= UStelescope(\underline{m}, \underline{n}) ⇒ |(us[\underline{m}] + (−uus[(\underline{m} + ((\underline{n} + 1) + 1))]))| <= UStelescope(\underline{m}, (\underline{n} + 1))⌉, p_{0}, c)]

[TelescopeNumerical(Indu) →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: |(us[\underline{m}] + (−uus[(\underline{m} + (\underline{n} + 1))]))| <= UStelescope(\underline{m}, \underline{n}) ⇒ |(us[\underline{m}] + (−uus[(\underline{m} + ((\underline{n} + 1) + 1))]))| <= UStelescope(\underline{m}, (\underline{n} + 1))]

[TelescopeNumerical(Indu) →^{tex} “TelescopeNumerical(Indu)”]

[TelescopeNumerical(Indu) →^{pyk} “lemma telescopeNumerical indu”]

TelescopeNumerical

[TelescopeNumerical →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: TelescopeNumerical(Base) ≫ |(us[\underline{m}] + (−uus[(\underline{m} + (0 + 1))]))| <= UStelescope(\underline{m}, 0); TelescopeNumerical(Indu) ≫ |(us[\underline{m}] + (−uus[(\underline{m} + (\underline{n} + 1))]))| <= UStelescope(\underline{m}, \underline{n}) ⇒ |(us[\underline{m}] + (−uus[(\underline{m} + ((\underline{n} + 1) + 1))]))| <= UStelescope(\underline{m}, (\underline{n} + 1)); Induction ⊳ |(us[\underline{m}] + (−uus[(\underline{m} + (0 + 1))]))| <= UStelescope(\underline{m}, 0) ⊳ |(us[\underline{m}] + (−uus[(\underline{m} + (\underline{n} + 1))]))| <= UStelescope(\underline{m}, \underline{n}) ⇒ |(us[\underline{m}] + (−uus[(\underline{m} + ((\underline{n} + 1) + 1))]))| <= UStelescope(\underline{m}, (\underline{n} + 1)) ≫ |(us[\underline{m}] + (−uus[(\underline{m} + (\underline{n} + 1))]))| <= UStelescope(\underline{m}, \underline{n})⌉, p_{0}, c)]

[TelescopeNumerical →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: |(us[\underline{m}] + (−uus[(\underline{m} + (\underline{n} + 1))]))| <= UStelescope(\underline{m}, \underline{n})]

[TelescopeNumerical →^{tex} “TelescopeNumerical”]

[TelescopeNumerical →^{pyk} “lemma telescopeNumerical”]

(+1)IsPositive(N)

[(+1)IsPositive(N) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: Nat(\underline{m}) ⊢⊢ Nonnegative(N) ⊳ ⊳ Nat(\underline{m}) ≫ 0 <= \underline{m}; Leq + 1 ⊳ 0 <= \underline{m} ≫ ˙¬ (0 <= (\underline{m} + 1) ⇒ ˙¬ ( ˙¬ (0 = (\underline{m} + 1))n)n)n⌉, p_{0}, c)]

[(+1)IsPositive(N) →^{stmt} SystemQ ⊢ ∀\underline{m}: Nat(\underline{m}) ⊢⊢ ˙¬ (0 <= (\underline{m} + 1) ⇒ ˙¬ ( ˙¬ (0 = (\underline{m} + 1))n)n)n]

[(+1)IsPositive(N) →^{tex} “(+1)IsPositive(N)”]

[(+1)IsPositive(N) →^{pyk} “lemma +1IsPositive(N)”]

DistributionOut(Minus)

[DistributionOut(Minus) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: Times(−1)Left ≫ ((−u1)∗(\underline{x}∗\underline{z})) = (−u(\underline{x}∗\underline{z})); eqSymmetry⊳((−u1)∗ (\underline{x} ∗ \underline{z})) = (−u(\underline{x} ∗ \underline{z})) ≫ (−u(\underline{x} ∗ \underline{z})) = ((−u1) ∗ (\underline{x} ∗ \underline{z})); timesCommutativity ≫ ((−u1) ∗ (\underline{x} ∗ \underline{z})) = ((\underline{x} ∗ \underline{z}) ∗ (−u1)); timesAssociativity ≫ ((\underline{x} ∗ \underline{z}) ∗ (−u1)) = (\underline{x} ∗ (\underline{z} ∗ (−u1))); Times(−1) ≫ (\underline{z} ∗ (−u1)) = (−u\underline{z}); EqMultiplicationLeft ⊳ (\underline{z} ∗ (−u1)) = (−u\underline{z}) ≫ (\underline{x} ∗ (\underline{z} ∗ (−u1))) = (\underline{x} ∗ (−u\underline{z})); eqTransitivity5 ⊳ (−u(\underline{x} ∗ \underline{z})) = ((−u1) ∗ (\underline{x} ∗ \underline{z})) ⊳ ((−u1) ∗ (\underline{x} ∗ \underline{z})) = ((\underline{x} ∗ \underline{z}) ∗ (−u1)) ⊳ ((\underline{x} ∗ \underline{z}) ∗ (−u1)) = (\underline{x} ∗ (\underline{z} ∗ (−u1))) ⊳ (\underline{x} ∗ (\underline{z} ∗ (−u1))) = (\underline{x} ∗ (−u\underline{z})) ≫ (−u(\underline{x} ∗ \underline{z})) = (\underline{x} ∗ (−u\underline{z})); EqAdditionLeft ⊳ (−u(\underline{x} ∗ \underline{z})) = (\underline{x} ∗ (−u\underline{z})) ≫ ((\underline{x} ∗ \underline{y}) + (−u(\underline{x} ∗ \underline{z}))) = ((\underline{x} ∗ \underline{y}) + (\underline{x} ∗ (−u\underline{z}))); DistributionOut ≫ ((\underline{x} ∗ \underline{y}) + (\underline{x} ∗ (−u\underline{z}))) = (\underline{x} ∗ (\underline{y} + (−u\underline{z}))); eqTransitivity ⊳ ((\underline{x} ∗ \underline{y}) + (−u(\underline{x} ∗ \underline{z}))) = ((\underline{x} ∗ \underline{y}) + (\underline{x} ∗ (−u\underline{z}))) ⊳ ((\underline{x} ∗ \underline{y}) + (\underline{x} ∗ (−u\underline{z}))) = (\underline{x} ∗ (\underline{y} + (−u\underline{z}))) ≫ ((\underline{x} ∗ \underline{y}) + (−u(\underline{x} ∗ \underline{z}))) = (\underline{x} ∗ (\underline{y} + (−u\underline{z})))⌉, p_{0}, c)]

[DistributionOut(Minus) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ((\underline{x} ∗ \underline{y}) + (−u(\underline{x} ∗ \underline{z}))) = (\underline{x} ∗ (\underline{y} + (−u\underline{z})))]

[DistributionOut(Minus) →^{tex} “DistributionOut(Minus)”]

[DistributionOut(Minus) →^{pyk} “lemma distributionOut(Minus)”]

PositiveToRight(Eq)(1term)

[PositiveToRight(Eq)(1term) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} = \underline{y} ⊢ eqSymmetry ⊳ \underline{x} = \underline{y} ≫ \underline{y} = \underline{x}; PositiveToLeft(Eq)(1term) ⊳ \underline{y} = \underline{x} ≫ (\underline{y} + (−u\underline{x})) = 0; eqSymmetry ⊳ (\underline{y} + (−u\underline{x})) = 0 ≫ 0 = (\underline{y} + (−u\underline{x}))⌉, p_{0}, c)]

[PositiveToRight(Eq)(1term) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} = \underline{y} ⊢ 0 = (\underline{y} + (−u\underline{x}))] [PositiveToRight(Eq)(1term) →^{tex} “PositiveToRight(Eq)(1 term)”]

[PositiveToRight(Eq)(1term) →^{pyk} “lemma positiveToRight(Eq)(1 term)”]

SameSeries(NumDiff)

[SameSeries(NumDiff) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{o}: ∀\underline{p}: ∀\underline{(n1)}: ∀\underline{(n2)}: \underline{o} = \underline{p} ⊢ \underline{(n1)} = \underline{(n2)} ⊢ SameSeries ⊳ \underline{o} = \underline{p} ≫ \underline{(fx)}[\underline{o}] = \underline{(fx)}[\underline{p}]; SameSeries ⊳ \underline{(n1)} = \underline{(n2)} ≫ \underline{(fy)}[\underline{(n1)}] = \underline{(fy)}[\underline{(n2)}]; EqNegated ⊳ \underline{(fy)}[\underline{(n1)}] = \underline{(fy)}[\underline{(n2)}] ≫ (−u\underline{(fy)}[\underline{(n1)}]) = (−u\underline{(fy)}[\underline{(n2)}]); AddEquations ⊳ \underline{(fx)}[\underline{o}] = \underline{(fx)}[\underline{p}] ⊳ (−u\underline{(fy)}[\underline{(n1)}]) = (−u\underline{(fy)}[\underline{(n2)}]) ≫ (\underline{(fx)}[\underline{o}] + (−u\underline{(fy)}[\underline{(n1)}])) = (\underline{(fx)}[\underline{p}] + (−u\underline{(fy)}[\underline{(n2)}])); SameNumerical ⊳ (\underline{(fx)}[\underline{o}] + (−u\underline{(fy)}[\underline{(n1)}])) = (\underline{(fx)}[\underline{p}] + (−u\underline{(fy)}[\underline{(n2)}])) ≫ |(\underline{(fx)}[\underline{o}] + (−u\underline{(fy)}[\underline{(n1)}]))| = |(\underline{(fx)}[\underline{p}] + (−u\underline{(fy)}[\underline{(n2)}]))|⌉, p_{0}, c)]

[SameSeries(NumDiff) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{o}: ∀\underline{p}: ∀\underline{(n1)}: ∀\underline{(n2)}: \underline{o} = \underline{p} ⊢ \underline{(n1)} = \underline{(n2)} ⊢ |(\underline{(fx)}[\underline{o}] + (−u\underline{(fy)}[\underline{(n1)}]))| = |(\underline{(fx)}[\underline{p}] + (−u\underline{(fy)}[\underline{(n2)}]))|]

[SameSeries(NumDiff) →^{tex} “SameSeries(NumDiff)”]

[SameSeries(NumDiff) →^{pyk} “lemma sameSeries(NumDiff)”]

ToNegatedDoubleImply

[ToNegatedDoubleImply →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} ⊢ \underline{b} ⊢ ˙¬ (\underline{c})n ⊢ \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⊢ MP2 ⊳ \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⊳ \underline{a} ⊳ \underline{b} ≫ \underline{c}; FromContradiction ⊳ \underline{c} ⊳ ˙¬ (\underline{c})n ≫ ˙¬ (\underline{a} ⇒ \underline{b} ⇒ \underline{c})n; ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: Ded⊳∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} ⊢ \underline{b} ⊢ ˙¬ (\underline{c})n ⊢ \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⊢ ˙¬ (\underline{a} ⇒ \underline{b} ⇒ \underline{c})n ≫ \underline{a} ⇒ \underline{b} ⇒ ˙¬ (\underline{c})n ⇒ \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ ˙¬ (\underline{a} ⇒ \underline{b} ⇒ \underline{c})n; \underline{a} ⊢ \underline{b} ⊢ ˙¬ (\underline{c})n ⊢ MP3 ⊳ \underline{a} ⇒ \underline{b} ⇒ ˙¬ (\underline{c})n ⇒ \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ ˙¬ (\underline{a} ⇒ \underline{b} ⇒ \underline{c})n ⊳ \underline{a} ⊳ \underline{b} ⊳ ˙¬ (\underline{c})n ≫ \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ ˙¬ (\underline{a} ⇒ \underline{b} ⇒ \underline{c})n; prop lemma imply negation ⊳ \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ ˙¬ (\underline{a} ⇒ \underline{b} ⇒ \underline{c})n ≫ ˙¬ (\underline{a} ⇒ \underline{b} ⇒ \underline{c})n⌉, p_{0}, c)]

[ToNegatedDoubleImply →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} ⊢ \underline{b} ⊢ ˙¬ (\underline{c})n ⊢ ˙¬ (\underline{a} ⇒ \underline{b} ⇒ \underline{c})n]

[ToNegatedDoubleImply →^{tex} “ToNegatedDoubleImply”]

[ToNegatedDoubleImply →^{pyk} “prop lemma to negated double imply”]

AddNegatedAll

[AddNegatedAll →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{a}: ∀\underline{b}: \underline{b} ⇒ \underline{a} ⊢ AddAll ⊳ \underline{b} ⇒ \underline{a} ≫ ∀_{obj}\underline{(v1)}: \underline{b} ⇒ ∀_{obj}\underline{(v1)}: \underline{a}; Contrapositive ⊳ ∀_{obj}\underline{(v1)}: \underline{b} ⇒ ∀_{obj}\underline{(v1)}: \underline{a} ≫ ˙¬ (∀_{obj}\underline{(v1)}: \underline{a})n ⇒ ˙¬ (∀_{obj}\underline{(v1)}: \underline{b})n⌉, p_{0}, c)]

[AddNegatedAll →^{stmt} SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{a}: ∀\underline{b}: \underline{b} ⇒ \underline{a} ⊢ ˙¬ (∀_{obj}\underline{(v1)}: \underline{a})n ⇒ ˙¬ (∀_{obj}\underline{(v1)}: \underline{b})n]

[AddNegatedAll →^{tex} “AddNegatedAll”]

[AddNegatedAll →^{pyk} “pred lemma addNegatedAll”]

(A)to( E )(Imply)

[(A)to( E )(Imply) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{a}: ∀_{obj}\underline{(v1)}: \underline{a} ⊢ A4 @ \underline{(v1)} ⊳ ∀_{obj}\underline{(v1)}: \underline{a} ≫ \underline{a}; AddDoubleNeg ⊳ \underline{a} ≫ ˙¬ ( ˙¬ (\underline{a})n)n; Gen ⊳ ˙¬ ( ˙¬ (\underline{a})n)n ≫ ∀_{obj}\underline{(v1)}: ˙¬ ( ˙¬ (\underline{a})n)n; AddDoubleNeg ⊳ ∀_{obj}\underline{(v1)}: ˙¬ ( ˙¬ (\underline{a})n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ ( ˙¬ (\underline{a})n)n)n)n; Repetition ⊳ ˙¬ ( ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ ( ˙¬ (\underline{a})n)n)n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ ( ˙¬ (\underline{a})n)n)n)n; ∀\underline{(v1)}: ∀\underline{a}: Ded ⊳ ∀\underline{(v1)}: ∀\underline{a}: ∀_{obj}\underline{(v1)}: \underline{a} ⊢ ˙¬ ( ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ ( ˙¬ (\underline{a})n)n)n)n ≫ ∀_{obj}\underline{(v1)}: \underline{a} ⇒ ˙¬ ( ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ ( ˙¬ (\underline{a})n)n)n)n⌉, p_{0}, c)]

[(A)to( E )(Imply) →^{stmt} SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{a}: ∀_{obj}\underline{(v1)}: \underline{a} ⇒ ˙¬ ( ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ ( ˙¬ (\underline{a})n)n)n)n]

[(A)to( E )(Imply) →^{tex} “(A)to(˜ E˜ )(Imply)”]

[(A)to( E )(Imply) →^{pyk} “pred lemma (A)to(˜ E˜ )(Imply)”]

(E)to( A )(Imply)

[(E)to( A )(Imply) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{a}: AutoImply ≫ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{a})n)n ⇒ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{a})n)n; Repetition ⊳ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{a})n)n ⇒ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{a})n)n ≫ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{a})n)n ⇒ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{a})n)n⌉, p_{0}, c)]

[(E)to( A )(Imply) →^{stmt} SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{a}: ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{a})n)n ⇒ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{a})n)n]

[(E)to( A )(Imply) →^{tex} “(E)to(˜ A˜ )(Imply)”]

[(E)to( A )(Imply) →^{pyk} “pred lemma (E)to(˜ A˜ )(Imply)”]

(E )to( A)(Imply)

[(E )to( A)(Imply) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{a}: ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ ( ˙¬ (\underline{a})n)n)n ⊢ AddDoubleNeg ⊳ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ ( ˙¬ (\underline{a})n)n)n ≫ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ ( ˙¬ (\underline{a})n)n)n)n)n; (A)to( E )(Imply) ≫ ∀_{obj}\underline{(v1)}: \underline{a} ⇒ ˙¬ ( ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ ( ˙¬ (\underline{a})n)n)n)n; MT ⊳ ∀_{obj}\underline{(v1)}: \underline{a} ⇒ ˙¬ ( ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ ( ˙¬ (\underline{a})n)n)n)n ⊳ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ ( ˙¬ (\underline{a})n)n)n)n)n ≫ ˙¬ (∀_{obj}\underline{(v1)}: \underline{a})n; ∀\underline{(v1)}: ∀\underline{a}: Ded ⊳ ∀\underline{(v1)}: ∀\underline{a}: ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ ( ˙¬ (\underline{a})n)n)n ⊢ ˙¬ (∀_{obj}\underline{(v1)}: \underline{a})n ≫ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ ( ˙¬ (\underline{a})n)n)n ⇒ ˙¬ (∀_{obj}\underline{(v1)}: \underline{a})n⌉, p_{0}, c)]

[(E )to( A)(Imply) →^{stmt} SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{a}: ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ ( ˙¬ (\underline{a})n)n)n ⇒ ˙¬ (∀_{obj}\underline{(v1)}: \underline{a})n]

[(E )to( A)(Imply) →^{tex} “(E˜ )to(˜ A)(Imply)”]

[(E )to( A)(Imply) →^{pyk} “pred lemma (E˜ )to(˜ A)(Imply)”]

ToNegatedAEA

[ToNegatedAEA →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{(v3)}: ∀\underline{a}: ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ ( ˙¬ (\underline{a})n)n)n)n)n ⊢ (E )to( A)(Imply) ≫ ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ ( ˙¬ (\underline{a})n)n)n ⇒ ˙¬ (∀_{obj}\underline{(v3)}: \underline{a})n; AddNegatedAll ⊳ ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ ( ˙¬ (\underline{a})n)n)n ⇒ ˙¬ (∀_{obj}\underline{(v3)}: \underline{a})n ≫ ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: \underline{a})n)n ⇒ ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ ( ˙¬ (\underline{a})n)n)n)n; (E)to( A )(Imply) ≫ ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: \underline{a})n)n ⇒ ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: \underline{a})n)n; ImplyTransitivity ⊳ ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: \underline{a})n)n ⇒ ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: \underline{a})n)n ⊳ ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: \underline{a})n)n ⇒ ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ ( ˙¬ (\underline{a})n)n)n)n ≫ ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: \underline{a})n)n ⇒ ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ ( ˙¬ (\underline{a})n)n)n)n; AddNegatedAll ⊳ ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: \underline{a})n)n ⇒ ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ ( ˙¬ (\underline{a})n)n)n)n ≫ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ ( ˙¬ (\underline{a})n)n)n)n)n ⇒ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: \underline{a})n)n)n; (E)to( A )(Imply) ≫ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ ( ˙¬ (\underline{a})n)n)n)n)n ⇒ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ ( ˙¬ (\underline{a})n)n)n)n)n; ImplyTransitivity ⊳ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ ( ˙¬ (\underline{a})n)n)n)n)n ⇒ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ ( ˙¬ (\underline{a})n)n)n)n)n ⊳ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ ( ˙¬ (\underline{a})n)n)n)n)n ⇒ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: \underline{a})n)n)n ≫ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ ( ˙¬ (\underline{a})n)n)n)n)n ⇒ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: \underline{a})n)n)n; MP ⊳ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ ( ˙¬ (\underline{a})n)n)n)n)n ⇒ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: \underline{a})n)n)n ⊳ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ ( ˙¬ (\underline{a})n)n)n)n)n ≫ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: \underline{a})n)n)n⌉, p_{0}, c)]

[ToNegatedAEA →^{stmt} SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{(v3)}: ∀\underline{a}: ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ ( ˙¬ (\underline{a})n)n)n)n)n ⊢ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: \underline{a})n)n)n]

[ToNegatedAEA →^{tex} “ToNegatedAEA ”]

[ToNegatedAEA →^{pyk} “pred lemma toNegatedAEA”]

Three2threeTerms(R)

[Three2threeTerms(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: PlusCommutativity(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; Three2twoTerms(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}]+(−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; PlusAssociativity(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}]+(−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; ==Symmetry ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}]+(−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}]+(−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; eqTransitivity ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}]+(−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}]+(−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}]+(−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}]+(−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, p_{0}, c)]

[Three2threeTerms(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}]+(−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}]+(−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[Three2threeTerms(R) →^{tex} “Three2threeTerms(R)”]

[Three2threeTerms(R) →^{pyk} “lemma three2threeTerms(R)”]

LessNeq(F)(Helper)

[LessNeq(F)(Helper) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})))n)n ⊢ A4 @ \underline{n} ⊳ ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})))n)n ≫ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{n} ⇒ \underline{(fx)}[\underline{n}] <= (\underline{(fy)}[\underline{n}] + (−u\underline{(ϵ)})))n)n; FirstConjunct ⊳ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{n} ⇒ \underline{(fx)}[\underline{n}] <= (\underline{(fy)}[\underline{n}] + (−u\underline{(ϵ)})))n)n ≫ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n; SecondConjunct ⊳ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{n} ⇒ \underline{(fx)}[\underline{n}] <= (\underline{(fy)}[\underline{n}] + (−u\underline{(ϵ)})))n)n ≫ \underline{n} <= \underline{n} ⇒ \underline{(fx)}[\underline{n}] <= (\underline{(fy)}[\underline{n}] + (−u\underline{(ϵ)})); leqReflexivity ≫ \underline{n} <= \underline{n}; MP ⊳ \underline{n} <= \underline{n} ⇒ \underline{(fx)}[\underline{n}] <= (\underline{(fy)}[\underline{n}]+(−u\underline{(ϵ)}))⊳\underline{n} <= \underline{n} ≫ \underline{(fx)}[\underline{n}] <= (\underline{(fy)}[\underline{n}]+(−u\underline{(ϵ)})); SwitchTerms(x <= y − z) ⊳ \underline{(fx)}[\underline{n}] <= (\underline{(fy)}[\underline{n}] + (−u\underline{(ϵ)})) ≫ \underline{(ϵ)} <= (\underline{(fy)}[\underline{n}] + (−u\underline{(fx)}[\underline{n}])); x <= |x| ≫ (\underline{(fy)}[\underline{n}]+(−u\underline{(fx)}[\underline{n}])) <= |(\underline{(fy)}[\underline{n}]+(−u\underline{(fx)}[\underline{n}]))|; leqTransitivity⊳\underline{(ϵ)} <= (\underline{(fy)}[\underline{n}] + (−u\underline{(fx)}[\underline{n}])) ⊳ (\underline{(fy)}[\underline{n}] + (−u\underline{(fx)}[\underline{n}])) <= |(\underline{(fy)}[\underline{n}] + (−u\underline{(fx)}[\underline{n}]))| ≫ \underline{(ϵ)} <= |(\underline{(fy)}[\underline{n}]+(−u\underline{(fx)}[\underline{n}]))|; NumericalDifference ≫ |(\underline{(fy)}[\underline{n}]+(−u\underline{(fx)}[\underline{n}]))| = |(\underline{(fx)}[\underline{n}] + (−u\underline{(fy)}[\underline{n}]))|; subLeqRight ⊳ |(\underline{(fy)}[\underline{n}] + (−u\underline{(fx)}[\underline{n}]))| = |(\underline{(fx)}[\underline{n}] + (−u\underline{(fy)}[\underline{n}]))| ⊳ \underline{(ϵ)} <= |(\underline{(fy)}[\underline{n}] + (−u\underline{(fx)}[\underline{n}]))| ≫ \underline{(ϵ)} <= |(\underline{(fx)}[\underline{n}] + (−u\underline{(fy)}[\underline{n}]))|; toNotLess ⊳ \underline{(ϵ)} <= |(\underline{(fx)}[\underline{n}] + (−u\underline{(fy)}[\underline{n}]))| ≫ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{n}] + (−u\underline{(fy)}[\underline{n}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{n}] + (−u\underline{(fy)}[\underline{n}]))| = \underline{(ϵ)})n)n)n)n; ToNegatedDoubleImply ⊳ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⊳ \underline{n} <= \underline{n} ⊳ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{n}] + (−u\underline{(fy)}[\underline{n}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{n}] + (−u\underline{(fy)}[\underline{n}]))| = \underline{(ϵ)})n)n)n)n ≫ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{n} <= \underline{n} ⇒ ˙¬ (|(\underline{(fx)}[\underline{n}] + (−u\underline{(fy)}[\underline{n}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{n}] + (−u\underline{(fy)}[\underline{n}]))| = \underline{(ϵ)})n)n)n)n; ∀\underline{m}: ∀\underline{n}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: Ded ⊳ ∀\underline{m}: ∀\underline{n}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})))n)n ⊢ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{n} <= \underline{n} ⇒ ˙¬ (|(\underline{(fx)}[\underline{n}] + (−u\underline{(fy)}[\underline{n}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{n}] + (−u\underline{(fy)}[\underline{n}]))| = \underline{(ϵ)})n)n)n)n ≫ ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})))n)n ⇒ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{n} <= \underline{n} ⇒ ˙¬ (|(\underline{(fx)}[\underline{n}] + (−u\underline{(fy)}[\underline{n}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{n}] + (−u\underline{(fy)}[\underline{n}]))| = \underline{(ϵ)})n)n)n)n⌉, p_{0}, c)]

[LessNeq(F)(Helper) →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})))n)n ⇒ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{n} <= \underline{n} ⇒ ˙¬ (|(\underline{(fx)}[\underline{n}] + (−u\underline{(fy)}[\underline{n}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{n}] + (−u\underline{(fy)}[\underline{n}]))| = \underline{(ϵ)})n)n)n)n]

[LessNeq(F)(Helper) →^{tex} “LessNeq(F)(Helper)”]

[LessNeq(F)(Helper) →^{pyk} “lemma lessNeq(F) helper”]

LessNeq(F)

[LessNeq(F) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ Repetition ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n; Ded ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ (∀_{obj}\underline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})))n)n)n)n)n)n; LessNeq(F)(Helper) ≫ ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})))n)n ⇒ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{n} <= \underline{n} ⇒ ˙¬ (|(\underline{(fx)}[\underline{n}] + (−u\underline{(fy)}[\underline{n}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{n}] + (−u\underline{(fy)}[\underline{n}]))| = \underline{(ϵ)})n)n)n)n; TwiceExistMP ⊳ ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})))n)n ⇒ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{n} <= \underline{n} ⇒ ˙¬ (|(\underline{(fx)}[\underline{n}] + (−u\underline{(fy)}[\underline{n}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{n}] + (−u\underline{(fy)}[\underline{n}]))| = \underline{(ϵ)})n)n)n)n ⊳ ˙¬ (∀_{obj}\underline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{n} <= \underline{n} ⇒ ˙¬ (|(\underline{(fx)}[\underline{n}] + (−u\underline{(fy)}[\underline{n}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{n}] + (−u\underline{(fy)}[\underline{n}]))| = \underline{(ϵ)})n)n)n)n; IntroExist @ \underline{n} ⊳ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{n} <= \underline{n} ⇒ ˙¬ (|(\underline{(fx)}[\underline{n}] + (−u\underline{(fy)}[\underline{n}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{n}] + (−u\underline{(fy)}[\underline{n}]))| = \underline{(ϵ)})n)n)n)n ≫ ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{n} <= \underline{m} ⇒ ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| = \underline{(ϵ)})n)n)n)n)n)n; Gen ⊳ ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{n} <= \underline{m} ⇒ ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| = \underline{(ϵ)})n)n)n)n)n)n ≫ ∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{n} <= \underline{m} ⇒ ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| = \underline{(ϵ)})n)n)n)n)n)n; IntroExist @ \underline{(ϵ)} ⊳ ∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{n} <= \underline{m} ⇒ ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| = \underline{(ϵ)})n)n)n)n)n)n ≫ ˙¬ (∀_{obj}\underline{(ϵ)}: ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{n} <= \underline{m} ⇒ ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| = \underline{(ϵ)})n)n)n)n)n)n)n)n; ToNegatedAEA ⊳ ˙¬ (∀_{obj}\underline{(ϵ)}: ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{n} <= \underline{m} ⇒ ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| = \underline{(ϵ)})n)n)n)n)n)n)n)n ≫ ˙¬ (∀_{obj}\underline{(ϵ)}: ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{m}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{n} <= \underline{m} ⇒ ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| = \underline{(ϵ)})n)n)n)n)n)n; Ded ⊳ ˙¬ (∀_{obj}\underline{(ϵ)}: ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{m}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{n} <= \underline{m} ⇒ ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| = \underline{(ϵ)})n)n)n)n)n)n ≫ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n)n; Repetition ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n)n ≫ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n)n⌉, p_{0}, c)] [LessNeq(F) →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n)n]

[LessNeq(F) →^{tex} “LessNeq(F)”]

[LessNeq(F) →^{pyk} “lemma lessNeq(F)”]

LessNeq(R)

[LessNeq(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ Repetition ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n; LessNeq(F) ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n)n; To!! == ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n)n ≫ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n⌉, p_{0}, c)]

[LessNeq(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ ˙¬ ({ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n]

[LessNeq(R) →^{tex} “LessNeq(R)”]

[LessNeq(R) →^{pyk} “lemma lessNeq(R)”]

x = x + (y − y)(R)

[x = x + (y − y)(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: Plus0(R) ≫ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; ==Symmetry ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; Negative(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; == Symmetry ⊳ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; EqAdditionLeft(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; eqTransitivity ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, p_{0}, c)]

[x = x + (y − y)(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[x = x + (y − y)(R) →^{tex} “x=x+(y-y)(R)”]

[x = x + (y − y)(R) →^{pyk} “lemma x=x+(y-y)(R)”]

x = x + y − y(R)

[x = x + y − y(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: x = x + (y − y)(R) ≫ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; PlusAssociativity(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; ==Symmetry ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; eqTransitivity ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, p_{0}, c)]

[x = x + y − y(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[x = x + y − y(R) →^{tex} “x=x+y-y(R)”]

[x = x + y − y(R) →^{pyk} “lemma x=x+y-y(R)”]

SubtractEquations(R)

[SubtractEquations(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀\underline{(fu)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fu)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fu)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fu)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fu)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ EqAddition(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fu)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fu)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}]+\underline{(fu)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}]+\underline{(fu)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; Plus0Left(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; eqTransitivity ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fu)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fu)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fu)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fu)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; PositiveToRight(Eq)(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fu)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fu)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fu)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fu)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; ==Symmetry ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fu)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fu)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fu)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fu)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; EqAdditionLeft(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fu)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fu)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fu)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fu)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; PlusAssociativity(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}]+\underline{(fu)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}]+\underline{(fu)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fu)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fu)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; Plus0(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; eqTransitivity4 ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}]+\underline{(fu)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}]+\underline{(fu)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fu)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fu)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fu)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fu)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}]+\underline{(fu)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}]+\underline{(fu)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; x = x + y − y(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; eqTransitivity4 ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}]+\underline{(fu)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}]+\underline{(fu)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}]+\underline{(fu)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}]+\underline{(fu)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, p_{0}, c)]

[SubtractEquations(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀\underline{(fu)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fu)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fu)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fu)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fu)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[SubtractEquations(R) →^{tex} “SubtractEquations(R)”]

[SubtractEquations(R) →^{pyk} “lemma subtractEquations(R)”]

NeqAddition(R)

[NeqAddition(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ˙¬ (\{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⊢ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢==Reflexivity ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; SubtractEquations(R) ⊳ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; FromContradiction ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}]+(−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ≫ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n; ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: Ded ⊳ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⊢ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ ˙¬ ({ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ≫ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ ˙¬ ({ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n; ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⊢ MP ⊳ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ ˙¬ ({ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⊳ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ ˙¬ ({ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n; prop lemma imply negation ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ ˙¬ ({ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ≫ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n⌉, p_{0}, c)]

[NeqAddition(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⊢ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n]

[NeqAddition(R) →^{tex} “NeqAddition(R)”]

[NeqAddition(R) →^{pyk} “lemma neqAddition(R)”]

PositiveToRight(Less)(R)

[PositiveToRight(Less)(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] <= (\underline{(fz)}[\overline{m}]+(−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ lessAddition(R) ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n; x = x + y − y(R) ≫ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; ==Symmetry ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; SubLessLeft(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n⌉, p_{0}, c)]

[PositiveToRight(Less)(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n]

[PositiveToRight(Less)(R) →^{tex} “PositiveToRight(Less)(R)”]

[PositiveToRight(Less)(R) →^{pyk} “lemma positiveToRight(Less)(R)”]

PositiveToRight(Less)(1term)(R)

[PositiveToRight(Less)(1term)(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ Plus0Left(R) ≫ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; ==Symmetry ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; SubLessLeft(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n; PositiveToRight(Less)(R) ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n⌉, p_{0}, c)]

[PositiveToRight(Less)(1term)(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n]

[PositiveToRight(Less)(1term)(R) →^{tex} “PositiveToRight(Less)(1 term)(R)”]

[PositiveToRight(Less)(1term)(R) →^{pyk} “lemma positiveToRight(Less)(1 term)(R)”]

LeqNeqLess(R)

[LeqNeqLess(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ LeqAntisymmetry(R) ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; FromContradiction ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ≫ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n; ∀\underline{(fx)}: ∀\underline{(fy)}: Ded ⊳ ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n; ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⊢ MP2 ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n; prop lemma imply negation ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ≫ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n; ToLess(R) ⊳ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ≫ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n⌉, p_{0}, c)]

[LeqNeqLess(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⊢ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n]

[LeqNeqLess(R) →^{tex} “LeqNeqLess(R)”]

[LeqNeqLess(R) →^{pyk} “lemma leqNeqLess(R)”]

SubLeqLeft(R)

[SubLeqLeft(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢==Symmetry ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; lemma eqLeq(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; LeqTransitivity(R) ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, p_{0}, c)]

[SubLeqLeft(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[SubLeqLeft(R) →^{tex} “SubLeqLeft(R)”]

[SubLeqLeft(R) →^{pyk} “lemma subLeqLeft(R)”]

ToLeq(Advanced)(R)

[ToLeq(Advanced)(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fep)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ PositiveToRight(Less)(1term)(R) ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n; PlusCommutativity(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; Three2threeTerms(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}]+\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}]+\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; x = x + y − y(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}]+\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}]+\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; ==Symmetry ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}]+\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}]+\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}]+\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}]+\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; eqTransitivity4 ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}]+\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}]+\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}]+\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}]+\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; JoinConjuncts ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}]+(−u\overline{(ϵ)})))n)n)n)n)n)n⊳\{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⇒ ˙¬ ({ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n)n; IntroExist @\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⇒ ˙¬ ({ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n)n ≫ ˙¬ (∀_{obj}\underline{(fep)}: ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fep)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⇒ ˙¬ ({ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fep)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fep)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n)n)n)n; ∀\underline{(fep)}: ∀\underline{(fx)}: ∀\underline{(fy)}: Ded ⊳ ∀\underline{(fep)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ ˙¬ (∀_{obj}\underline{(fep)}: ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fep)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⇒ ˙¬ ({ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fep)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fep)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n)n)n)n ≫ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\underline{(fep)}: ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fep)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⇒ ˙¬ ({ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fep)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fep)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n)n)n)n; ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fep)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⇒ ˙¬ ({ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fep)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fep)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⊢ ToNegatedAnd ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fep)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⇒ ˙¬ ({ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fep)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fep)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ≫ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fep)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⇒ ˙¬ ({ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fep)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fep)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n)n)n; Gen ⊳ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fep)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⇒ ˙¬ ({ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fep)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fep)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n)n)n ≫ ∀_{obj}\underline{(fep)}: ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fep)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⇒ ˙¬ ({ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fep)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fep)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n)n)n; (A )to( E) ⊳ ∀_{obj}\underline{(fep)}: ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fep)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⇒ ˙¬ ({ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fep)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fep)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n)n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\underline{(fep)}: ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fep)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⇒ ˙¬ ({ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fep)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fep)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n)n)n)n)n; MT ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\underline{(fep)}: ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fep)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⇒ ˙¬ ({ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fep)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fep)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n)n)n)n ⊳ ˙¬ ( ˙¬ (∀_{obj}\underline{(fep)}: ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fep)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⇒ ˙¬ ({ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fep)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fep)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n)n)n)n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}]+(−u\overline{(ϵ)})))n)n)n)n)n)n)n; FromNotLess(R)⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, p_{0}, c)] [ToLeq(Advanced)(R) →^{stmt} SystemQ ⊢ ∀\underline{(fep)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fep)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⇒ ˙¬ ({ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fep)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fep)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[ToLeq(Advanced)(R) →^{tex} “ToLeq(Advanced)(R)”]

[ToLeq(Advanced)(R) →^{pyk} “lemma toLeq(Advanced)(R)”]

LeqLessTransitivity(R)

[LeqLessTransitivity(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ LessLeq(R) ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; LessNeq(R) ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n; SubLeqLeft(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fz)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; LeqAntisymmetry(R) ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fz)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; FromContradiction ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}]+(−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ≫ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n; ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: Ded ⊳ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n; ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ MP2 ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n; prop lemma imply negation ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}]+(−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ≫ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n; LessLeq(R) ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; LeqTransitivity(R) ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; LeqNeqLess(R)⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ≫ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n⌉, p_{0}, c)]

[LeqLessTransitivity(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n]

[LeqLessTransitivity(R) →^{tex} “LeqLessTransitivity(R)”] [LeqLessTransitivity(R) →^{pyk} “lemma leqLessTransitivity(R)”]

NegativeToLeft(Eq)(R)

[NegativeToLeft(Eq)(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ EqAddition(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; Three2threeTerms(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}]+\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}]+\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; x = x + y − y(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}]+\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}]+\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; ==Symmetry ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}]+\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}]+\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}]+\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}]+\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; eqTransitivity4 ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}]+\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}]+\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}]+\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}]+\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, p_{0}, c)] [NegativeToLeft(Eq)(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[NegativeToLeft(Eq)(R) →^{tex} “NegativeToLeft(Eq)(R)”]

[NegativeToLeft(Eq)(R) →^{pyk} “lemma negativeToLeft(Eq)(R)”]

NegativeToRight(Less)(R)

[NegativeToRight(Less)(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ lessAddition(R) ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] <= ({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n; Three2threeTerms(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; x = x + y − y(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; ==Symmetry ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; eqTransitivity ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; SubLessLeft(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] <= ({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n⌉, p_{0}, c)]

[NegativeToRight(Less)(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n]

[NegativeToRight(Less)(R) →^{tex} “NegativeToRight(Less)(R)”]

[NegativeToRight(Less)(R) →^{pyk} “lemma negativeToRight(Less)(R)”]

!! == Symmetry

[!! == Symmetry →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢==Symmetry ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; ∀\underline{(fx)}: ∀\underline{(fy)}: Ded ⊳ ∀\underline{(fx)}: ∀\underline{(fy)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⊢ MT ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}]+(−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ≫ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n⌉, p_{0}, c)]

[!! == Symmetry →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⊢ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n]

[!! == Symmetry →^{tex} “!!==Symmetry”]

[!! == Symmetry →^{pyk} “lemma !!==Symmetry”]

SwitchTerms(x <= y − z)

[SwitchTerms(x <= y − z) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= (\underline{y} + (−u\underline{z})) ⊢ negativeToLeft(Leq) ⊳ \underline{x} <= (\underline{y} + (−u\underline{z})) ≫ (\underline{x} + \underline{z}) <= \underline{y}; plusCommutativity ≫ (\underline{x} + \underline{z}) = (\underline{z} + \underline{x}); subLeqLeft ⊳ (\underline{x} + \underline{z}) = (\underline{z} + \underline{x}) ⊳ (\underline{x} + \underline{z}) <= \underline{y} ≫ (\underline{z} + \underline{x}) <= \underline{y}; PositiveToRight(Leq) ⊳ (\underline{z} + \underline{x}) <= \underline{y} ≫ \underline{z} <= (\underline{y} + (−u\underline{x}))⌉, p_{0}, c)] [SwitchTerms(x <= y − z) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= (\underline{y} + (−u\underline{z})) ⊢ \underline{z} <= (\underline{y} + (−u\underline{x}))]

[SwitchTerms(x <= y − z) →^{tex} “SwitchTerms(x<=y-z)”]

[SwitchTerms(x <= y − z) →^{pyk} “lemma switchTerms(x<=y-z)”]

Plus0Left(R)

[Plus0Left(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: Plus0(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; PlusCommutativity(R) ≫ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; eqTransitivity ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, p_{0}, c)]

[Plus0Left(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[Plus0Left(R) →^{tex} “Plus0Left(R)”]

[Plus0Left(R) →^{pyk} “lemma plus0Left(R)”]

PositiveToRight(Eq)(R)

[PositiveToRight(Eq)(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ EqAddition(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; x = x + y − y(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; eqTransitivity ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, p_{0}, c)]

[PositiveToRight(Eq)(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[PositiveToRight(Eq)(R) →^{tex} “PositiveToRight(Eq)(R)”]

[PositiveToRight(Eq)(R) →^{pyk} “lemma positiveToRight(Eq)(R)”]

EqAdditionLeft(R)

[EqAdditionLeft(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ EqAddition(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; PlusCommutativity(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; PlusCommutativity(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; eqTransitivity4 ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, p_{0}, c)]

[EqAdditionLeft(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[EqAdditionLeft(R) →^{tex} “EqAdditionLeft(R)”]

[EqAdditionLeft(R) →^{pyk} “lemma eqAdditionLeft(R)”]

Three2twoTerms(R)

[Three2twoTerms(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀\underline{(fu)}: \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fu)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fu)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ EqAdditionLeft(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fu)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fu)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fu)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fu)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; PlusAssociativity(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}]+(−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; eqTransitivity ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}]+(−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fu)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fu)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}]+(−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fu)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fu)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, p_{0}, c)]

[Three2twoTerms(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀\underline{(fu)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fu)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fu)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}]+(−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fu)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fu)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}] [Three2twoTerms(R) →^{tex} “Three2twoTerms(R)”]

[Three2twoTerms(R) →^{pyk} “lemma three2twoTerms(R)”]

To!! ==

[To!! == →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ From == ⊳\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n; ∀\underline{m}: ∀\underline{n}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: Ded ⊳ ∀\underline{m}: ∀\underline{n}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n; ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n)n ⊢ MT ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n)n ≫ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n⌉, p_{0}, c)]

[To!! == →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n)n ⊢ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n]

[To!! ==→^{tex} “To!!==”]

[To!! == →^{pyk} “lemma to!!==”]

PositiveToRight(Less)(1term)

[PositiveToRight(Less)(1term) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⊢ plus0Left ≫ (0 + \underline{x}) = \underline{x}; eqSymmetry ⊳ (0 + \underline{x}) = \underline{x} ≫ \underline{x} = (0 + \underline{x}); SubLessLeft ⊳ \underline{x} = (0 + \underline{x}) ⊳ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ≫ ˙¬ ((0 + \underline{x}) <= \underline{y} ⇒ ˙¬ ( ˙¬ ((0 + \underline{x}) = \underline{y})n)n)n; PositiveToRight(Less) ⊳ ˙¬ ((0 + \underline{x}) <= \underline{y} ⇒ ˙¬ ( ˙¬ ((0 + \underline{x}) = \underline{y})n)n)n ≫ ˙¬ (0 <= (\underline{y} + (−u\underline{x})) ⇒ ˙¬ ( ˙¬ (0 = (\underline{y} + (−u\underline{x})))n)n)n⌉, p_{0}, c)]

[PositiveToRight(Less)(1term) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⊢ ˙¬ (0 <= (\underline{y} + (−u\underline{x})) ⇒ ˙¬ ( ˙¬ (0 = (\underline{y} + (−u\underline{x})))n)n)n]

[PositiveToRight(Less)(1term) →^{tex} “PositiveToRight(Less)(1 term)”]

[PositiveToRight(Less)(1term) →^{pyk} “lemma positiveToRight(Less)(1 term)”]

(A )to( E)

[(A )to( E) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{a}: ∀_{obj}\underline{(v1)}: ˙¬ (\underline{a})n ⊢ AddDoubleNeg ⊳ ∀_{obj}\underline{(v1)}: ˙¬ (\underline{a})n ≫ ˙¬ ( ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{a})n)n)n; Repetition ⊳ ˙¬ ( ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{a})n)n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{a})n)n)n⌉, p_{0}, c)]

[(A )to( E) →^{stmt} SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{a}: ∀_{obj}\underline{(v1)}: ˙¬ (\underline{a})n ⊢ ˙¬ ( ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{a})n)n)n]

[(A )to( E) →^{tex} “(A˜ )to(˜ E)”]

[(A )to( E) →^{pyk} “pred lemma (A˜ )to(˜ E)”]

NegativeToRight(Eq)(R)

[NegativeToRight(Eq)(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ EqAddition(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; x = x + y − y(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; Three2threeTerms(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; eqTransitivity4 ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}]+\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]+\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, p_{0}, c)]

[NegativeToRight(Eq)(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[NegativeToRight(Eq)(R) →^{tex} “NegativeToRight(Eq)(R)”]

[NegativeToRight(Eq)(R) →^{pyk} “lemma negativeToRight(Eq)(R)”]

NegativeToRight(Eq)(1term)(R)

[NegativeToRight(Eq)(1term)(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ NegativeToRight(Eq)(R) ⊳ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; Plus0Left(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; eqTransitivity ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, p_{0}, c)]

[NegativeToRight(Eq)(1term)(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[NegativeToRight(Eq)(1term)(R) →^{tex} “NegativeToRight(Eq)(1 term)(R)”]

[NegativeToRight(Eq)(1term)(R) →^{pyk} “lemma negativeToRight(Eq)(1 term)(R)”]

DoubleMinus(R)

[DoubleMinus(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: Negative(R) ≫ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; PlusCommutativity(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; ==Symmetry ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; eqTransitivity ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; NegativeToRight(Eq)(1term)(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, p_{0}, c)]

[DoubleMinus(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}] [DoubleMinus(R) →^{tex} “DoubleMinus(R)”]

[DoubleMinus(R) →^{pyk} “lemma doubleMinus(R)”]

UniqueNegative(R)

[UniqueNegative(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ PlusCommutativity(R) ≫ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; eqTransitivity ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; PositiveToRight(Eq)(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; PlusCommutativity(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; eqTransitivity ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; PositiveToRight(Eq)(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; ==Symmetry ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; eqTransitivity ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, p_{0}, c)]

[UniqueNegative(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[UniqueNegative(R) →^{tex} “UniqueNegative(R)”]

[UniqueNegative(R) →^{pyk} “lemma uniqueNegative(R)”]

SubtractEquationsLeft(R)

[SubtractEquationsLeft(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀\underline{(fu)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fu)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fu)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ PlusCommutativity(R) ≫ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; PlusCommutativity(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fu)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fu)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fu)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fu)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; eqTransitivity4 ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fu)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fu)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fu)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fu)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fu)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fu)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fu)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fu)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; SubtractEquations(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fz)}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fu)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fu)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fu)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fu)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, p_{0}, c)]

[SubtractEquationsLeft(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀\underline{(fu)}: \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fu)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fu)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fu)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fu)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[SubtractEquationsLeft(R) →^{tex} “SubtractEquationsLeft(R)”]

[SubtractEquationsLeft(R) →^{pyk} “lemma subtractEquationsLeft(R)”]

EqNegated(R)

[EqNegated(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ Negative(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; Negative(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; == Symmetry ⊳ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; eqTransitivity ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; SubtractEquationsLeft(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, p_{0}, c)]

[EqNegated(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[EqNegated(R) →^{tex} “EqNegated(R)”]

[EqNegated(R) →^{pyk} “lemma eqNegated(R)”]

NeqNegated(R)

[NeqNegated(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⊢ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ EqNegated(R) ⊳ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; DoubleMinus(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; ==Symmetry ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; DoubleMinus(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; eqTransitivity4 ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; FromContradiction ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}]+(−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ≫ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n; ∀\underline{(fx)}: ∀\underline{(fy)}: Ded ⊳ ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⊢ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ ˙¬ ({ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ≫ ˙¬ ({ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ ˙¬ ({ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n; ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⊢ MP ⊳ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ ˙¬ ({ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⊳ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ ˙¬ ({ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n; prop lemma imply negation ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ ˙¬ ({ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ≫ ˙¬ ({ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n⌉, p_{0}, c)]

[NeqNegated(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⊢ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n]

[NeqNegated(R) →^{tex} “NeqNegated(R)”]

[NeqNegated(R) →^{pyk} “lemma neqNegated(R)”]

−0 = 0(R)

[−0 = 0(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ Negative(R) ≫ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; Plus0(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; UniqueNegative(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, p_{0}, c)]

[−0 = 0(R) →^{stmt} SystemQ ⊢ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[−0 = 0(R) →^{tex} “-0=0(R)”]

[−0 = 0(R) →^{pyk} “lemma -0=0(R)”]

NegativeNegated(R)

[NegativeNegated(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ LessNegated(R) ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n; −0 = 0(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; SubLessLeft(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n⌉, p_{0}, c)]

[NegativeNegated(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n]

[NegativeNegated(R) →^{tex} “NegativeNegated(R)”]

[NegativeNegated(R) →^{pyk} “lemma negativeNegated(R)”]

FromLeqGeq(R)

[FromLeqGeq(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ \underline{a} ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ \underline{a} ⊢ LeqTotality(R) ≫ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; FromDisjuncts ⊳ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ \underline{a} ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ \underline{a} ≫ \underline{a}⌉, p_{0}, c)]

[FromLeqGeq(R) →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ \underline{a} ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ \underline{a} ⊢ \underline{a}]

[FromLeqGeq(R) →^{tex} “FromLeqGeq(R)”]

[FromLeqGeq(R) →^{pyk} “lemma from leqGeq(R)”]

0 <= |x|(R)

[0 <= |x|(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ NonnegativeNumerical(R) ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; ==Symmetry ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; SubLeqRight(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (|f\underline{(fx)}|[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; ∀\underline{(fx)}: ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⊢ ToLess(R) ⊳ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ≫ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n; NegativeNumerical(R) ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; == Symmetry ⊳ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; NegativeNegated(R) ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n; LessLeq(R) ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; SubLeqRight(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (|f\underline{(fx)}|[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; ∀\underline{(fx)}: Ded ⊳ ∀\underline{(fx)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (|f\underline{(fx)}|[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (|f\underline{(fx)}|[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; Ded ⊳ ∀\underline{(fx)}: ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (|f\underline{(fx)}|[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (|f\underline{(fx)}|[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; FromNegations ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (|f\underline{(fx)}|[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (|f\underline{(fx)}|[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (|f\underline{(fx)}|[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, p_{0}, c)]

[0 <= |x|(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (|f\underline{(fx)}|[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[0 <= |x|(R) →^{tex} “0<=|x|(R)”]

[0 <= |x|(R) →^{pyk} “lemma 0<=|x|(R)”]

PositiveNegated(R)

[PositiveNegated(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ LessNegated(R) ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n; −0 = 0(R) ≫ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; SubLessRight(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n⌉, p_{0}, c)]

[PositiveNegated(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n]

[PositiveNegated(R) →^{tex} “PositiveNegated(R)”]

[PositiveNegated(R) →^{pyk} “lemma positiveNegated(R)”]

AddEquations(R)

[AddEquations(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀\underline{(fu)}: \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fu)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fu)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ EqAddition(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; EqAdditionLeft(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fu)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fu)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fu)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fu)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; eqTransitivity ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fu)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fu)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fu)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fu)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, p_{0}, c)]

[AddEquations(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀\underline{(fu)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fu)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fu)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fu)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fu)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[AddEquations(R) →^{tex} “AddEquations(R)”]

[AddEquations(R) →^{pyk} “lemma addEquations(R)”]

Times(−1)(R)

[Times(−1)(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: Negative(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; PlusCommutativity(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; eqTransitivity ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; EqMultiplicationLeft(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; x ∗ 0 = 0(R)(fff) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; eqTransitivity ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; Distribution(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; ==Symmetry ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; eqTransitivity ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; PositiveToRight(Eq)(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; Plus0Left(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; eqTransitivity ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; Times1(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; EqNegated(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; eqTransitivity ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, p_{0}, c)]

[Times(−1)(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[Times(−1)(R) →^{tex} “Times(-1)(R)”]

[Times(−1)(R) →^{pyk} “lemma times(-1)(R)”]

Times(−1)Left(R)

[Times(−1)Left(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: Times(−1)(R) ≫ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; TimesCommutativity(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; eqTransitivity ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, p_{0}, c)]

[Times(−1)Left(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[Times(−1)Left(R) →^{tex} “Times(-1)Left(R)”]

[Times(−1)Left(R) →^{pyk} “lemma times(-1)Left(R)”]

−x − y = −(x + y)(R)

[−x − y = −(x + y)(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: Times(−1)Left(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; Times(−1)Left(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; AddEquations(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; ==Symmetry ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; DistributionOut(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; Times(−1)Left(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; eqTransitivity4 ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, p_{0}, c)]

[−x − y = −(x + y)(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[−x − y = −(x + y)(R) →^{tex} “-x-y=-(x+y)(R)”]

[−x − y = −(x + y)(R) →^{pyk} “lemma -x-y=-(x+y)(R)”]

LessTotality(R)

[LessTotality(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⊢ ˙¬ ({ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⊢ FromNotLess(R) ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; !! == Symmetry ⊳ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}]+(−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ≫ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n; LeqNeqLess(R) ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ≫ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n; ∀\underline{(fx)}: ∀\underline{(fy)}: Ded ⊳ ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⊢ ˙¬ ({ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⊢ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ˙¬ (\{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⇒ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n; Repetition ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ˙¬ (\{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⇒ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ˙¬ (\{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⇒ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n⌉, p_{0}, c)]

[LessTotality(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ˙¬ ({ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⇒ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n]

[LessTotality(R) →^{tex} “LessTotality(R)”]

[LessTotality(R) →^{pyk} “lemma lessTotality(R)”]

SameNumerical(R)

[SameNumerical(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ NonnegativeNumerical(R) ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; SubLeqRight(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; NonnegativeNumerical(R) ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fy)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fy)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; ==Symmetry ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fy)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fy)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fy)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fy)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; eqTransitivity4 ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fy)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fy)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fy)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fy)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⊢ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ ToLess(R) ⊳ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ≫ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n; NegativeNumerical(R) ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; SubLessLeft(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n; NegativeNumerical(R) ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fy)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fy)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; == Symmetry ⊳ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fy)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fy)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fy)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fy)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; EqNegated(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; eqTransitivity4 ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fy)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fy)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fy)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fy)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; ∀\underline{(fx)}: ∀\underline{(fy)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ Ded ⊳ ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fy)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fy)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fy)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fy)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; Ded ⊳ ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⊢ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fy)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fy)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fy)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fy)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; FromNegations ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fy)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fy)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fy)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fy)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fy)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fy)}|[\overline{m}]+(−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; MP⊳\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fy)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fy)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fy)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fy)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, p_{0}, c)]

[SameNumerical(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fy)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fy)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[SameNumerical(R) →^{tex} “SameNumerical(R)”]

[SameNumerical(R) →^{pyk} “lemma sameNumerical(R)”]

MinusNegated(R)

[MinusNegated(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: DoubleMinus(R) ≫ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; EqAddition(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; ==Symmetry ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; −x − y = −(x + y)(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; PlusCommutativity(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; EqNegated(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; eqTransitivity4 ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; ==Symmetry ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, p_{0}, c)]

[MinusNegated(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[MinusNegated(R) →^{tex} “MinusNegated(R)”]

[MinusNegated(R) →^{pyk} “lemma minusNegated(R)”]

PositiveNumerical(R)

[PositiveNumerical(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ LessLeq(R) ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; NonnegativeNumerical(R) ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, p_{0}, c)] [PositiveNumerical(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[PositiveNumerical(R) →^{tex} “PositiveNumerical(R)”]

[PositiveNumerical(R) →^{pyk} “lemma positiveNumerical(R)”]

SignNumerical(+)(R)

[SignNumerical(+)(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ PositiveNumerical(R) ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; PositiveNegated(R) ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n; NegativeNumerical(R) ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; DoubleMinus(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; eqTransitivity ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; ==Symmetry ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; eqTransitivity ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, p_{0}, c)]

[SignNumerical(+)(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}] [SignNumerical(+)(R) →^{tex} “SignNumerical(+)(R)”]

[SignNumerical(+)(R) →^{pyk} “lemma signNumerical(+)(R)”]

NonnegativeNumerical(R)

[NonnegativeNumerical(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ from <<== ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n; NonnegativeNumerical(F) ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n ≫ |f\underline{(fx)}| = \underline{(fx)}; (Adgic)SameR ⊳ |f\underline{(fx)}| = \underline{(fx)} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; Repetition ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, p_{0}, c)]

[NonnegativeNumerical(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[NonnegativeNumerical(R) →^{tex} “NonnegativeNumerical(R)”]

[NonnegativeNumerical(R) →^{pyk} “lemma nonnegativeNumerical(R)”]

NegativeNumerical(R)

[NegativeNumerical(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n ⊢ FromLess(R) ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n; to <<== ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; FromContradiction ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ≫ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n)n; ∀\underline{(fx)}: Ded ⊳ ∀\underline{(fx)}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n ⊢ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n)n ≫ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n ⇒ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n)n; ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ MP ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n ⇒ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n)n ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n ⇒ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n)n; prop lemma imply negation ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n ⇒ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n)n ≫ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n)n; NegativeNumerical(F) ≫ |f\underline{(fx)}| = \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}; (Adgic)SameR ⊳ |f\underline{(fx)}| = {ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; Repetition ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, p_{0}, c)]

[NegativeNumerical(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[NegativeNumerical(R) →^{tex} “NegativeNumerical(R)”]

[NegativeNumerical(R) →^{pyk} “lemma negativeNumerical(R)”]

LeqNegated(R)

[LeqNegated(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ leqAddition(R) ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; Negative(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; SubLeqLeft(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; PlusCommutativity(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; SubLeqRight(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; leqAddition(R) ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; Plus0Left(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; x = x + y − y(R) ≫ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; ==Symmetry ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; SubLeqLeft(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; SubLeqRight(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, p_{0}, c)]

[LeqNegated(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[LeqNegated(R) →^{tex} “LeqNegated(R)”]

[LeqNegated(R) →^{pyk} “lemma leqNegated(R)”]

LessNegated(R)

[LessNegated(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ LessLeq(R) ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; LeqNegated(R)⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; LessNeq(R) ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n; NeqNegated(R) ⊳ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}]+(−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ≫ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n; !! == Symmetry ⊳ ˙¬ ({ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ≫ ˙¬ ({ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n; LeqNeqLess(R) ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ≫ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n⌉, p_{0}, c)]

[LessNegated(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n]

[LessNegated(R) →^{tex} “LessNegated(R)”]

[LessNegated(R) →^{pyk} “lemma lessNegated(R)”]

SubLeqRight(R)

[SubLeqRight(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fz)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ lemma eqLeq(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; LeqTransitivity(R) ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fz)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fz)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, p_{0}, c)]

[SubLeqRight(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fz)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fz)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fz)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[SubLeqRight(R) →^{tex} “SubLeqRight(R)”]

[SubLeqRight(R) →^{pyk} “lemma subLeqRight(R)”]

FromLess(R)

[FromLess(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ LessLeq(R) ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; LeqAntisymmetry(R) ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; LessNeq(R) ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n; FromContradiction⊳\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ≫ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n; ∀\underline{(fx)}: ∀\underline{(fy)}: Ded ⊳ ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ≫ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n; ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ MP ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n; prop lemma imply negation ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ≫ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n⌉, p_{0}, c)]

[FromLess(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n]

[FromLess(R) →^{tex} “FromLess(R)”]

[FromLess(R) →^{pyk} “lemma fromLess(R)”]

DistributionOut(R)

[DistributionOut(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: Distribution(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; ==Symmetry ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, p_{0}, c)] [DistributionOut(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[DistributionOut(R) →^{tex} “DistributionOut(R)”]

[DistributionOut(R) →^{pyk} “lemma distributionOut(R)”]

x ∗ 0 + x = x(R)

[x ∗ 0 + x = x(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: Times1(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; ==Symmetry ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; EqAdditionLeft(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; Distribution(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; ==Symmetry ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; Plus0Left(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; EqMultiplicationLeft(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; eqTransitivity5 ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 1\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, p_{0}, c)]

[x ∗ 0 + x = x(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[x ∗ 0 + x = x(R) →^{tex} “x∗0+x=x(R)”]

[x ∗ 0 + x = x(R) →^{pyk} “lemma x∗0+x=x(R)”]

x ∗ 0 = 0(R)(fff)

[x ∗ 0 = 0(R)(fff) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: x = x + (y − y)(R) ≫ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; PlusAssociativity(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; ==Symmetry ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; x ∗ 0 + x = x(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; EqAddition(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; Negative(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; eqTransitivity5 ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] + \underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, p_{0}, c)]

[x ∗ 0 = 0(R)(fff) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[x ∗ 0 = 0(R)(fff) →^{tex} “x∗0=0(R)(fff)”]

[x ∗ 0 = 0(R)(fff) →^{pyk} “lemma x∗0=0(R)fff”]

SignNumerical(R)

[SignNumerical(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ NegativeNegated(R) ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n; SignNumerical(+)(R) ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f{ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; DoubleMinus(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; SameNumerical(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f{ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; eqTransitivity ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f{ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f{ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; ==Symmetry ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; ∀\underline{(fx)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ EqNegated(R) ⊳ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; −0 = 0(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; == Symmetry ⊳ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; eqTransitivity4 ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; ==Symmetry ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; SameNumerical(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; ∀\underline{(fx)}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ SignNumerical(+)(R) ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; ∀\underline{(fx)}: Ded ⊳ ∀\underline{(fx)}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⇒ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; Ded ⊳ ∀\underline{(fx)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; Ded ⊳ ∀\underline{(fx)}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; LessTotality(R) ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ˙¬ (\{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⇒ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n; From3Disjuncts ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ˙¬ (\{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\})n ⇒ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⇒ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, p_{0}, c)]

[SignNumerical(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[SignNumerical(R) →^{tex} “SignNumerical(R)”]

[SignNumerical(R) →^{pyk} “lemma signNumerical(R)”]

NumericalDifference(R)

[NumericalDifference(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: SignNumerical(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f{ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}|[\overline{m}]+(−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; MinusNegated(R) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; SameNumerical(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}|[\overline{m}]+(−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f{ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; eqTransitivity ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f{ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}|[\overline{m}]+(−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}|[\overline{m}]+(−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f{ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f{ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f{ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, p_{0}, c)]

[NumericalDifference(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f{ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f{ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (d_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] + \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}[\underline{m}])\}\})n)n\}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[NumericalDifference(R) →^{tex} “NumericalDifference(R)”]

[NumericalDifference(R) →^{pyk} “lemma numericalDifference(R)”]

x <= |x|(R)

[x <= |x|(R) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ NonnegativeNumerical(R) ≫ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; ==Symmetry ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; lemma eqLeq(R) ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (|f\underline{(fx)}|[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; ∀\underline{(fx)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ 0 <= |x|(R) ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (|f\underline{(fx)}|[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; LeqTransitivity(R) ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (|f\underline{(fx)}|[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (|f\underline{(fx)}|[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; ∀\underline{(fx)}: Ded ⊳ ∀\underline{(fx)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (|f\underline{(fx)}|[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (|f\underline{(fx)}|[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; Ded ⊳ ∀\underline{(fx)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (|f\underline{(fx)}|[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (|f\underline{(fx)}|[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; FromLeqGeq(R) ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (|f\underline{(fx)}|[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (|f\underline{(fx)}|[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (|f\underline{(fx)}|[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, p_{0}, c)]

[x <= |x|(R) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (|f\underline{(fx)}|[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(|f\underline{(fx)}|[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}] [x <= |x|(R) →^{tex} “x<=|x|(R)”] [x <= |x|(R) →^{pyk} “lemma x<=|x|(R)”]

USlimitIsUpperBound(Helper)

[USlimitIsUpperBound(Helper) →^{tex} “USlimitIsUpperBound(Helper)”] [USlimitIsUpperBound(Helper) →^{pyk} “lemma USlimitIsUpperBound helper”]

USlimitIsUpperBound

[USlimitIsUpperBound →^{tex} “USlimitIsUpperBound”] [USlimitIsUpperBound →^{pyk} “lemma USlimitIsUpperBound”]

(−1) ∗ (−1) + (−1) ∗ 1 = 0(R)

[(−1) ∗ (−1) + (−1) ∗ 1 = 0(R) →^{tex} “(-1)∗(-1)+(-1)∗1=0(R)”]

[(−1) ∗ (−1) + (−1) ∗ 1 = 0(R) →^{pyk} “lemma (-1)∗(-1)+(-1)∗1=0(R)”]

(−1) ∗ (−1) = 1(R)

[(−1) ∗ (−1) = 1(R) →^{tex} “(-1)∗(-1)=1(R)”]

[(−1) ∗ (−1) = 1(R) →^{pyk} “lemma (-1)∗(-1)=1(R)”]

0 < 1Helper(R)

[0 < 1Helper(R) →^{tex} “0<1Helper(R)”] [0 < 1Helper(R) →^{pyk} “lemma 0<1Helper(R)”]

0 < 1(R)

[0 < 1(R) →^{tex} “0<1(R)”]

[0 < 1(R) →^{pyk} “lemma 0<1(R)”]

ExpZero(Exact)(R)

[ExpZero(Exact)(R) →^{tex} “ExpZero(Exact)(R)”] [ExpZero(Exact)(R) →^{pyk} “lemma expZero exact(R)”]

PositiveBase(R)(Base)

[PositiveBase(R)(Base) →^{tex} “PositiveBase(R)(Base)”] [PositiveBase(R)(Base) →^{pyk} “lemma positiveBase(R) base”]

Three2twoFactors(R)

[Three2twoFactors(R) →^{tex} “Three2twoFactors(R)”] [Three2twoFactors(R) →^{pyk} “lemma three2twoFactors(R)”]

x = x ∗ y ∗ (1/y)(R)

[x = x ∗ y ∗ (1/y)(R) →^{tex} “x=x∗y∗(1/y)(R)”] [x = x ∗ y ∗ (1/y)(R) →^{pyk} “lemma x=x∗y∗(1/y)(R)”]

NeqMultiplication(R)

[NeqMultiplication(R) →^{tex} “NeqMultiplication(R)”] [NeqMultiplication(R) →^{pyk} “lemma neqMultiplication(R)”]

LessTransitivity(R)

[LessTransitivity(R) →^{tex} “LessTransitivity(R)”] [LessTransitivity(R) →^{pyk} “lemma lessTransitivity(R)”]

0 < 2(R)

[0 < 2(R) →^{tex} “0<2(R)”]

[0 < 2(R) →^{pyk} “lemma 0<2(R)”]

SameExp(R)(Base)

[SameExp(R)(Base) →^{tex} “SameExp(R)(Base)”] [SameExp(R)(Base) →^{pyk} “lemma sameExp(R) base”]

SameExp(R)(Indu)

[SameExp(R)(Indu) →^{tex} “SameExp(R)(Indu)”] [SameExp(R)(Indu) →^{pyk} “lemma sameExp(R) indu”]

SameExp(R)

[SameExp(R) →^{tex} “SameExp(R)”] [SameExp(R) →^{pyk} “lemma sameExp(R)”]

SubNeqLeft(R)

[SubNeqLeft(R) →^{tex} “SubNeqLeft(R)”] [SubNeqLeft(R) →^{pyk} “lemma subNeqLeft(R)”]

SubNeqRight(R)

[SubNeqRight(R) →^{tex} “SubNeqRight(R)”] [SubNeqRight(R) →^{pyk} “lemma subNeqRight(R)”]

NonzeroFactors(R)

[NonzeroFactors(R) →^{tex} “NonzeroFactors(R)”] [NonzeroFactors(R) →^{pyk} “lemma nonzeroFactors(R)”]

NonnegativeFactors(R)

[NonnegativeFactors(R) →^{tex} “NonnegativeFactors(R)”] [NonnegativeFactors(R) →^{pyk} “lemma nonnegativeFactors(R)”]

PositiveFactors(R)

[PositiveFactors(R) →^{tex} “PositiveFactors(R)”] [PositiveFactors(R) →^{pyk} “lemma positiveFactors(R)”]

LessDivision(R)

[LessDivision(R) →^{tex} “LessDivision(R)”] [LessDivision(R) →^{pyk} “lemma lessDivision(R)”]

0 < 1/2(R)

[0 < 1/2(R) →^{tex} “0<1/2(R)”]

[0 < 1/2(R) →^{pyk} “lemma 0<1/2(R)”]

PositiveToRight(Eq)(1term)(R)

[PositiveToRight(Eq)(1term)(R) →^{tex} “PositiveToRight(Eq)(1 term)(R)”] [PositiveToRight(Eq)(1term)(R) →^{pyk} “lemma positiveToRight(Eq)(1 term)(R)”]

Exp(+1)(R)

[Exp(+1)(R) →^{tex} “Exp(+1)(R)”] [Exp(+1)(R) →^{pyk} “lemma exp(+1)(R)”]

PositiveBase(R)(Indu)

[PositiveBase(R)(Indu) →^{tex} “PositiveBase(R)(Indu)”] [PositiveBase(R)(Indu) →^{pyk} “lemma positiveBase(R) indu”]

PositiveBase(R)

[PositiveBase(R) →^{tex} “PositiveBase(R)”] [PositiveBase(R) →^{pyk} “lemma positiveBase(R)”]

−x ∗ y = −(x ∗ y)(R)

[−x ∗ y = −(x ∗ y)(R) →^{tex} “-x∗y=-(x∗y)(R)”] [−x ∗ y = −(x ∗ y)(R) →^{pyk} “lemma -x∗y=-(x∗y)(R)”]

PositiveToLeft(Eq)(R)

[PositiveToLeft(Eq)(R) →^{tex} “PositiveToLeft(Eq)(R)”] [PositiveToLeft(Eq)(R) →^{pyk} “lemma positiveToLeft(Eq)(R)”]

Times1Left(R)

[Times1Left(R) →^{tex} “Times1Left(R)”] [Times1Left(R) →^{pyk} “lemma times1Left(R)”]

x + x = 2 ∗ x(R)

[x + x = 2 ∗ x(R) →^{tex} “x+x=2∗x(R)”] [x + x = 2 ∗ x(R) →^{pyk} “lemma x+x=2∗x(R)”]

(1/2)x + (1/2)x = x(R)

[(1/2)x + (1/2)x = x(R) →^{tex} “(1/2)x+(1/2)x=x(R)”] [(1/2)x + (1/2)x = x(R) →^{pyk} “lemma (1/2)x+(1/2)x=x(R)”]

DistributionOut(Minus)(R)

[DistributionOut(Minus)(R) →^{tex} “DistributionOut(Minus)(R)”] [DistributionOut(Minus)(R) →^{pyk} “lemma distributionOut(Minus)(R)”]

(1/2)(x + y) − x = (1/2)(y − x)(R)

[(1/2)(x + y) − x = (1/2)(y − x)(R) →^{tex} “(1/2)(x+y)-x=(1/2)(y-x)(R)”] [(1/2)(x + y) − x = (1/2)(y − x)(R) →^{pyk} “lemma (1/2)(x+y)-x=(1/2)(y-x)(R)”]

IntervalSize(R)(Base)

[IntervalSize(R)(Base) →^{tex} “IntervalSize(R)(Base)”] [IntervalSize(R)(Base) →^{pyk} “lemma intervalSize(R) base”]

LessMultiplicationLeft(R)

[LessMultiplicationLeft(R) →^{tex} “LessMultiplicationLeft(R)”] [LessMultiplicationLeft(R) →^{pyk} “lemma lessMultiplicationLeft(R)”]

NegativeToLeft(Less)(R)

[NegativeToLeft(Less)(R) →^{tex} “NegativeToLeft(Less)(R)”] [NegativeToLeft(Less)(R) →^{pyk} “lemma negativeToLeft(Less)(R)”]

NegativeToLeft(Less)(1term)(R)

[NegativeToLeft(Less)(1term)(R) →^{tex} “NegativeToLeft(Less)(1 term)(R)”] [NegativeToLeft(Less)(1term)(R) →^{pyk} “lemma negativeToLeft(Less)(1 term)(R)”]

y − (1/2)(x + y) = (1/2)(y − x)(R)

[y − (1/2)(x + y) = (1/2)(y − x)(R) →^{tex} “y-(1/2)(x+y)=(1/2)(y-x)(R)”] [y − (1/2)(x + y) = (1/2)(y − x)(R) →^{pyk} “lemma y-(1/2)(x+y)=(1/2)(y-x)(R)”]

IntervalSize(R)(Indu)

[IntervalSize(R)(Indu) →^{tex} “IntervalSize(R)(Indu)”] [IntervalSize(R)(Indu) →^{pyk} “lemma intervalSize(R) indu”]

IntervalSize(R)

[IntervalSize(R) →^{tex} “IntervalSize(R)”] [IntervalSize(R) →^{pyk} “lemma intervalSize(R)”]

XSlessUS(R)

[XSlessUS(R) →^{tex} “XSlessUS(R)”] [XSlessUS(R) →^{pyk} “lemma XSlessUS(R)”]

USdecreasing(+1)(R)

[USdecreasing(+1)(R) →^{tex} “USdecreasing(+1)(R)”] [USdecreasing(+1)(R) →^{pyk} “lemma USdecreasing(+1)(R)”]

ExpUnbounded(Base)

[ExpUnbounded(Base) →^{tex} “ExpUnbounded(Base)”] [ExpUnbounded(Base) →^{pyk} “lemma expUnbounded base”]

ExpUnbounded(Indu)

[ExpUnbounded(Indu) →^{tex} “ExpUnbounded(Indu)”] [ExpUnbounded(Indu) →^{pyk} “lemma expUnbounded indu”]

ExpUnbounded

[ExpUnbounded →^{tex} “ExpUnbounded”] [ExpUnbounded →^{pyk} “lemma expUnbounded”]

1 <= x + 1(N)

[1 <= x + 1(N) →^{tex} “1<=x+1(N)”]

[1 <= x + 1(N) →^{pyk} “lemma 1<=x+1(N)”]

ExpNonzero(Base)

[ExpNonzero(Base) →^{tex} “ExpNonzero(Base)”] [ExpNonzero(Base) →^{pyk} “lemma expNonzero base”]

ExpNonzero(Indu)

[ExpNonzero(Indu) →^{tex} “ExpNonzero(Indu)”] [ExpNonzero(Indu) →^{pyk} “lemma expNonzero indu”]

ExpNonzero

[ExpNonzero →^{tex} “ExpNonzero”] [ExpNonzero →^{pyk} “lemma expNonzero”]

ExpNonzero(2)

[ExpNonzero(2) →^{tex} “ExpNonzero(2)”] [ExpNonzero(2) →^{pyk} “lemma expNonzero(2)”]

HalfBase(Base)

[HalfBase(Base) →^{tex} “HalfBase(Base)”] [HalfBase(Base) →^{pyk} “lemma halfBase base”]

HalfBase(Indu)

[HalfBase(Indu) →^{tex} “HalfBase(Indu)”] [HalfBase(Indu) →^{pyk} “lemma halfBase indu”]

MultiplyEquations(R)

[MultiplyEquations(R) →^{tex} “MultiplyEquations(R)”] [MultiplyEquations(R) →^{pyk} “lemma multiplyEquations(R)”]

NonreciprocalToRight(Eq)(1term)(R)

[NonreciprocalToRight(Eq)(1term)(R) →^{tex} “NonreciprocalToRight(Eq)(1 term)(R)”] [NonreciprocalToRight(Eq)(1term)(R) →^{pyk} “lemma nonreciprocalToRight(Eq)(1 term)(R)”]

PositiveNonzero(R)

[PositiveNonzero(R) →^{tex} “PositiveNonzero(R)”] [PositiveNonzero(R) →^{pyk} “lemma positiveNonzero(R)”]

NonzeroProduct(2)(R)

[NonzeroProduct(2)(R) →^{tex} “NonzeroProduct(2)(R)”] [NonzeroProduct(2)(R) →^{pyk} “lemma nonzeroProduct(2)(R)”]

HalfBase

[HalfBase →^{tex} “HalfBase”] [HalfBase →^{pyk} “lemma halfBase”]

Three2threeFactors(R)

[Three2threeFactors(R) →^{tex} “Three2threeFactors(R)”] [Three2threeFactors(R) →^{pyk} “lemma three2threeFactors(R)”]

x ∗ y = zBackwards(R)

[x ∗ y = zBackwards(R) →^{tex} “x∗y=zBackwards(R)”] [x ∗ y = zBackwards(R) →^{pyk} “lemma x∗y=zBackwards(R)”]

PositiveInverted(R)

[PositiveInverted(R) →^{tex} “PositiveInverted(R)”] [PositiveInverted(R) →^{pyk} “lemma positiveInverted(R)”]

ReciprocalToRight(Less)(R)

[ReciprocalToRight(Less)(R) →^{tex} “ReciprocalToRight(Less)(R)”] [ReciprocalToRight(Less)(R) →^{pyk} “lemma reciprocalToRight(Less)(R)”]

ReciprocalToRight(Less)(1term)(R)

[ReciprocalToRight(Less)(1term)(R) →^{tex} “ReciprocalToRight(Less)(1 term)(R)”] [ReciprocalToRight(Less)(1term)(R) →^{pyk} “lemma reciprocalToRight(Less)(1 term)(R)”]

NonreciprocalToLeft(Less)(R)

[NonreciprocalToLeft(Less)(R) →^{tex} “NonreciprocalToLeft(Less)(R)”] [NonreciprocalToLeft(Less)(R) →^{pyk} “lemma nonreciprocalToLeft(Less)(R)”]

1 < x ∗ y(R)

[1 < x ∗ y(R) →^{tex} “1<x∗y(R)”] [1 < x ∗ y(R) →^{pyk} “lemma 1<x∗y(R)”]

SwitchFactors(1/x < y)(R)

[SwitchFactors(1/x < y)(R) →^{tex} “SwitchFactors(1/x<y)(R)”] [SwitchFactors(1/x < y)(R) →^{pyk} “lemma switchFactors(1/x<y)(R)”]

SmallHalving

[SmallHalving →^{tex} “SmallHalving”] [SmallHalving →^{pyk} “lemma smallHalving”]

IntervalSize(anyPositive)

[IntervalSize(anyPositive) →^{tex} “IntervalSize(anyPositive)”] [IntervalSize(anyPositive) →^{pyk} “lemma intervalSize(anyPositive)”]

USdecreasing(+n)(Base)

[USdecreasing(+n)(Base) →^{tex} “USdecreasing(+n)(Base)”] [USdecreasing(+n)(Base) →^{pyk} “lemma USdecreasing(+n) base”]

USdecreasing(+n)(Indu)

[USdecreasing(+n)(Indu) →^{tex} “USdecreasing(+n)(Indu)”] [USdecreasing(+n)(Indu) →^{pyk} “lemma USdecreasing(+n) indu”]

USdecreasing(+n)

[USdecreasing(+n) →^{tex} “USdecreasing(+n)”] [USdecreasing(+n) →^{pyk} “lemma USdecreasing(+n)”]

USdecreasing

[USdecreasing →^{tex} “USdecreasing”] [USdecreasing →^{pyk} “lemma USdecreasing”]

LeqAdditionLeft(R)

[LeqAdditionLeft(R) →^{tex} “LeqAdditionLeft(R)”] [LeqAdditionLeft(R) →^{pyk} “lemma leqAdditionLeft(R)”]

ToNotLess(R)

[ToNotLess(R) →^{tex} “ToNotLess(R)”] [ToNotLess(R) →^{pyk} “lemma toNotLess(R)”]

LimitOfUSIsLeq

[LimitOfUSIsLeq →^{tex} “LimitOfUSIsLeq”] [LimitOfUSIsLeq →^{pyk} “lemma limitOfUSIsLeq”]

SubtractEquations(Less)(R)

[SubtractEquations(Less)(R) →^{tex} “SubtractEquations(Less)(R)”] [SubtractEquations(Less)(R) →^{pyk} “lemma subtractEquations(Less)(R)”]

SubtractEquationsLeft(Less)(R)

[SubtractEquationsLeft(Less)(R) →^{tex} “SubtractEquationsLeft(Less)(R)”] [SubtractEquationsLeft(Less)(R) →^{pyk} “lemma subtractEquationsLeft(Less)(R)”]

LessNegated(Negative)(R)

[LessNegated(Negative)(R) →^{tex} “LessNegated(Negative)(R)”]

[LessNegated(Negative)(R) →^{pyk} “lemma lessNegated(Negative)(R)”]

FromNegatedAnd(Imply)

[FromNegatedAnd(Imply) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ ( ˙¬ (\underline{a} ⇒ ˙¬ (\underline{b})n)n)n ⊢ \underline{a} ⊢ FromNegatedAnd ⊳ ˙¬ ( ˙¬ (\underline{a} ⇒ ˙¬ (\underline{b})n)n)n ⊳ \underline{a} ≫ ˙¬ (\underline{b})n; ∀\underline{a}: ∀\underline{b}: Ded ⊳ ∀\underline{a}: ∀\underline{b}: ˙¬ ( ˙¬ (\underline{a} ⇒ ˙¬ (\underline{b})n)n)n ⊢ \underline{a} ⊢ ˙¬ (\underline{b})n ≫ ˙¬ ( ˙¬ (\underline{a} ⇒ ˙¬ (\underline{b})n)n)n ⇒ \underline{a} ⇒ ˙¬ (\underline{b})n⌉, p_{0}, c)]

[FromNegatedAnd(Imply) →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ ( ˙¬ (\underline{a} ⇒ ˙¬ (\underline{b})n)n)n ⇒ \underline{a} ⇒ ˙¬ (\underline{b})n]

[FromNegatedAnd(Imply) →^{tex} “FromNegatedAnd(Imply)”]

[FromNegatedAnd(Imply) →^{pyk} “prop lemma from negated and (imply)”]

RemoveDoubleNeg(Consequent)

[RemoveDoubleNeg(Consequent) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ ˙¬ ( ˙¬ (\underline{b})n)n ⊢ \underline{a} ⊢ MP ⊳ \underline{a} ⇒ ˙¬ ( ˙¬ (\underline{b})n)n ⊳ \underline{a} ≫ ˙¬ ( ˙¬ (\underline{b})n)n; RemoveDoubleNeg ⊳ ˙¬ ( ˙¬ (\underline{b})n)n ≫ \underline{b}; ∀\underline{a}: ∀\underline{b}: Ded ⊳ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ ˙¬ ( ˙¬ (\underline{b})n)n ⊢ \underline{a} ⊢ \underline{b} ≫ \underline{a} ⇒ ˙¬ ( ˙¬ (\underline{b})n)n ⇒ \underline{a} ⇒ \underline{b}; \underline{a} ⇒ ˙¬ ( ˙¬ (\underline{b})n)n ⊢ MP ⊳ \underline{a} ⇒ ˙¬ ( ˙¬ (\underline{b})n)n ⇒ \underline{a} ⇒ \underline{b} ⊳ \underline{a} ⇒ ˙¬ ( ˙¬ (\underline{b})n)n ≫ \underline{a} ⇒ \underline{b}⌉, p_{0}, c)]

[RemoveDoubleNeg(Consequent) →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ ˙¬ ( ˙¬ (\underline{b})n)n ⊢ \underline{a} ⇒ \underline{b}]

[RemoveDoubleNeg(Consequent) →^{tex} “RemoveDoubleNeg(Consequent)”]

[RemoveDoubleNeg(Consequent) →^{pyk} “prop lemma remove double neg (consequent)”]

FromNotUpperBound

[FromNotUpperBound →^{tex} “FromNotUpperBound”]

[FromNotUpperBound →^{pyk} “lemma fromNotUpperBound”]

LeqNUB

[LeqNUB →^{tex} “LeqNUB”]

[LeqNUB →^{pyk} “lemma leqNUB”]

USlimitIsLeastUpperBound(Helper)

[USlimitIsLeastUpperBound(Helper) →^{tex} “USlimitIsLeastUpperBound(Helper)”]

[USlimitIsLeastUpperBound(Helper) →^{pyk} “lemma USlimitIsLeastUpperBound helper”]

USlimitIsLeastUpperBound

[USlimitIsLeastUpperBound →^{tex} “USlimitIsLeastUpperBound”]

[USlimitIsLeastUpperBound →^{pyk} “lemma USlimitIsLeastUpperBound”]

ExistMP3

[ExistMP3 →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{(v3)}: ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{d} ⊢ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{a})n)n ⊢ ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (\underline{b})n)n ⊢ ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ (\underline{c})n)n ⊢ ExistMP2 ⊳ \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{d} ⊳ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{a})n)n ⊳ ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (\underline{b})n)n ≫ \underline{c} ⇒ \underline{d}; ExistMP ⊳ \underline{c} ⇒ \underline{d} ⊳ ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ (\underline{c})n)n ≫ \underline{d}⌉, p_{0}, c)]

[ExistMP3 →^{stmt} SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{(v3)}: ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{d} ⊢ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{a})n)n ⊢ ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (\underline{b})n)n ⊢ ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ (\underline{c})n)n ⊢ \underline{d}]

[ExistMP3 →^{tex} “ExistMP3”]

[ExistMP3 →^{pyk} “pred lemma exist mp3”]

GreaterPositive(N)

[GreaterPositive(N) →^{tex} “GreaterPositive(N)”]

[GreaterPositive(N) →^{pyk} “lemma greaterPositive(N)”]

ysFClose(Helper)

[ysFClose(Helper) →^{tex} “ysFClose(Helper)”]

[ysFClose(Helper) →^{pyk} “lemma ysFClose helper”]

ysFClose

[ysFClose →^{tex} “ysFClose”]

[ysFClose →^{pyk} “lemma ysFClose”]

ysFCauchy(Helper)

[ysFCauchy(Helper) →^{tex} “ysFCauchy(Helper)”]

[ysFCauchy(Helper) →^{pyk} “lemma ysFCauchy helper”]

ysFCauchy

[ysFCauchy →^{tex} “ysFCauchy”]

[ysFCauchy →^{pyk} “lemma ysFCauchy”]

from <<==

[from <<== →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ Repetition ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; Repetition ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ WeakenOr2 ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n; ∀\underline{(fx)}: ∀\underline{(fy)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ From ==≫ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n; WeakenOr1 ⊳ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n; Ded ⊳ ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n ≫ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n; Ded ⊳ ∀\underline{(fx)}: ∀\underline{(fy)}: \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n; FromDisjuncts ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n; Repetition ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n⌉, p_{0}, c)]

[from <<== →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n]

[from <<==→^{tex} “from<<==”]

[from <<== →^{pyk} “lemma from<<==”]

to <<==

[to <<== →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n ⊢ Repetition ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n; ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ Repetition ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n; WeakenOr2 ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; ∀\underline{(fx)}: ∀\underline{(fy)}: ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n ⊢ To == ⊳∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; WeakenOr1 ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; Ded ⊳ ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; Ded ⊳ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; FromDisjuncts ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ⊳ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; Repetition ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}⌉, p_{0}, c)]

[to <<== →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

[to <<==→^{tex} “to<<==”]

[to <<== →^{pyk} “lemma to<<==”]

NonnegativeNumerical(F)

[NonnegativeNumerical(F) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n ⊢ NumericalF ≫ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n ⇒ |f\underline{(fx)}| = \underline{(fx)} ⇒ ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n)n ⇒ |f\underline{(fx)}| = \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\})n)n; FirstConjunct ⊳ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n ⇒ |f\underline{(fx)}| = \underline{(fx)} ⇒ ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n)n ⇒ |f\underline{(fx)}| = \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\})n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n ⇒ |f\underline{(fx)}| = \underline{(fx)}; MP ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n ⇒ |f\underline{(fx)}| = \underline{(fx)} ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n ≫ |f\underline{(fx)}| = \underline{(fx)}⌉, p_{0}, c)]

[NonnegativeNumerical(F) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n ⊢ |f\underline{(fx)}| = \underline{(fx)}]

[NonnegativeNumerical(F) →^{tex} “NonnegativeNumerical(F)”]

[NonnegativeNumerical(F) →^{pyk} “lemma nonnegativeNumerical(F)”]

NegativeNumerical(F)

[NegativeNumerical(F) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n)n ⊢ NumericalF ≫ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n ⇒ |f\underline{(fx)}| = \underline{(fx)} ⇒ ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n)n ⇒ |f\underline{(fx)}| = \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\})n)n; SecondConjunct ⊳ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n ⇒ |f\underline{(fx)}| = \underline{(fx)} ⇒ ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n)n ⇒ |f\underline{(fx)}| = \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\})n)n ≫ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n)n ⇒ |f\underline{(fx)}| = \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}; MP ⊳ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n)n ⇒ |f\underline{(fx)}| = \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\} ⊳ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n)n ≫ |f\underline{(fx)}| = \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}⌉, p_{0}, c)]

[NegativeNumerical(F) →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n ⇒ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] + (−u\underline{(fx)}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n)n ⊢ |f\underline{(fx)}| = \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (−u\underline{(fx)}[\underline{m}])\}\})n)n\}] [NegativeNumerical(F) →^{tex} “NegativeNumerical(F)”] [NegativeNumerical(F) →^{pyk} “lemma negativeNumerical(F)”]

tester1

[tester1 →^{pyk} “tester1”]

tester2

[tester2 →^{pyk} “tester2”]

tester3

[tester3 →^{pyk} “tester3”]

tester4

[tester4 →^{pyk} “tester4”]

tester5

[tester5 →^{pyk} “tester5”]

tester6

[tester6 →^{pyk} “tester6”] The pyk compiler, version 0.grue.20060417+ by Klaus Grue GRD-2006-12-29.UTC:10:12:14.905583 = MJD-54098.TAI:10:12:47.905583 = LGT-4674103967905583e-6