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Nat(∗), ⟨∗≡ ∗ | ∗:==∗⟩_{Me}, ⟨∗≡^{1} ∗ | ∗:==∗⟩_{Me}, ⟨∗≡^{∗} ∗ | ∗:==∗⟩_{Me}, lemma eqLeq(R), ∗(exp)∗, 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), Tester1, Tester2, Tester3, Tester4, Tester5, Tester6,

Nat(∗)

[Nat(x) →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[Nat(x) ¨= λc.⌈x⌉ ∈_{t} (⌈V_{2n}⌉ : : ⌈M⌉ : : ⌈N ⌉ : : T)]⌉)]

⟨∗≡ ∗ | ∗:==∗⟩_{Me}

[⟨a≡b|x:==t⟩_{Me} →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[⟨a≡b|x:==t⟩_{Me} ¨= ⟨⌈a⌉≡^{1}⌈b⌉|⌈x⌉:==⌈t⌉⟩_{Me}]⌉)]

⟨∗≡^{1} ∗ | ∗:==∗⟩_{Me}

[⟨a≡^{1}b|x:==t⟩_{Me} →^{val} a!x!t! If(If(b =^{r} ⌈∀_{obj}u: v⌉, b^{1} =^{t} x, F), a =^{t} b, If(b =^{t} x, a =^{t} t, If( a =^{r} b, ⟨a^{t}≡^{∗}b^{t}|x:==t⟩_{Me}, F)))]

⟨∗≡^{∗} ∗ | ∗:==∗⟩_{Me}

[⟨a≡^{∗}b|x:==t⟩_{Me} →^{val} b!x!t!If(a, T, If(⟨a^{h}≡^{1}b^{h}|x:==t⟩_{Me}, ⟨a^{t}≡^{∗}b^{t}|x:==t⟩_{Me}, F))]

lemma eqLeq(R)

[lemma eqLeq(R) →^{tex} “eqLeq(R)”]

∗(exp)∗

[x(exp)y →^{tex} “(#1. (expARGH!) #2. )”]

sup

[sup →^{prio}

Preassociative [sup], [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)], [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)]; 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] ∗], [∗\\∗]; ]

[sup →^{tex} “sup”]

[sup →^{pyk} “sup”]

ToNegatedAnd(1)

[ToNegatedAnd(1) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ (\underline{a})n ⊢ \underline{a} ⊢ FromContradiction ⊳ \underline{a} ⊳ ˙¬ (\underline{a})n ≫ ˙¬ (\underline{b})n; ∀\underline{a}: ∀\underline{b}: Ded ⊳ ∀\underline{a}: ∀\underline{b}: ˙¬ (\underline{a})n ⊢ \underline{a} ⊢ ˙¬ (\underline{b})n ≫ ˙¬ (\underline{a})n ⇒ \underline{a} ⇒ ˙¬ (\underline{b})n; ˙¬ (\underline{a})n ⊢ MP ⊳ ˙¬ (\underline{a})n ⇒ \underline{a} ⇒ ˙¬ (\underline{b})n ⊳ ˙¬ (\underline{a})n ≫ \underline{a} ⇒ ˙¬ (\underline{b})n; ToNegatedAnd ⊳ \underline{a} ⇒ ˙¬ (\underline{b})n ≫ ˙¬ ( ˙¬ (\underline{a} ⇒ ˙¬ (\underline{b})n)n)n⌉, p_{0}, c)]

[ToNegatedAnd(1) →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{b}: ˙¬ (\underline{a})n ⊢ ˙¬ ( ˙¬ (\underline{a} ⇒ ˙¬ (\underline{b})n)n)n]

[ToNegatedAnd(1) →^{tex} “ToNegatedAnd(1)”]

[ToNegatedAnd(1) →^{pyk} “prop lemma to negated and(1)”]

UniqueNegative

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

[UniqueNegative →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: (\underline{x} + \underline{y}) = 0 ⊢ (\underline{x} + \underline{z}) = 0 ⊢ \underline{y} = \underline{z}]

[UniqueNegative →^{tex} “UniqueNegative”]

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

DoubleMinus

[DoubleMinus →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: Negative ≫ ((−u\underline{x}) + (−u(−u\underline{x}))) = 0; x + y = zBackwards ⊳ ((−u\underline{x}) + (−u(−u\underline{x}))) = 0 ≫ 0 = ((−u(−u\underline{x})) + (−u\underline{x})); NegativeToLeft(Eq) ⊳ 0 = ((−u(−u\underline{x})) + (−u\underline{x})) ≫ (0 + \underline{x}) = (−u(−u\underline{x})); plus0Left ≫ (0 + \underline{x}) = \underline{x}; Equality ⊳ (0 + \underline{x}) = (−u(−u\underline{x})) ⊳ (0 + \underline{x}) = \underline{x} ≫ (−u(−u\underline{x})) = \underline{x}⌉, p_{0}, c)]

[DoubleMinus →^{stmt} SystemQ ⊢ ∀\underline{x}: (−u(−u\underline{x})) = \underline{x}]

[DoubleMinus →^{tex} “DoubleMinus”]

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

MinusNegated

[MinusNegated →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: DoubleMinus ≫ (−u(−u\underline{y})) = \underline{y}; eqAddition ⊳ (−u(−u\underline{y})) = \underline{y} ≫ ((−u(−u\underline{y})) + (−u\underline{x})) = (\underline{y} + (−u\underline{x})); eqSymmetry ⊳ ((−u(−u\underline{y})) + (−u\underline{x})) = (\underline{y} + (−u\underline{x})) ≫ (\underline{y} + (−u\underline{x})) = ((−u(−u\underline{y})) + (−u\underline{x})); −x − y = −(x + y) ≫ ((−u(−u\underline{y})) + (−u\underline{x})) = (−u((−u\underline{y}) + \underline{x})); plusCommutativity ≫ ((−u\underline{y}) + \underline{x}) = (\underline{x} + (−u\underline{y})); EqNegated ⊳ ((−u\underline{y}) + \underline{x}) = (\underline{x} + (−u\underline{y})) ≫ (−u((−u\underline{y}) + \underline{x})) = (−u(\underline{x} + (−u\underline{y}))); eqTransitivity4 ⊳ (\underline{y} + (−u\underline{x})) = ((−u(−u\underline{y})) + (−u\underline{x})) ⊳ ((−u(−u\underline{y})) + (−u\underline{x})) = (−u((−u\underline{y}) + \underline{x})) ⊳ (−u((−u\underline{y}) + \underline{x})) = (−u(\underline{x}+(−u\underline{y}))) ≫ (\underline{y} +(−u\underline{x})) = (−u(\underline{x}+(−u\underline{y}))); eqSymmetry ⊳(\underline{y} +(−u\underline{x})) = (−u(\underline{x} + (−u\underline{y}))) ≫ (−u(\underline{x} + (−u\underline{y}))) = (\underline{y} + (−u\underline{x}))⌉, p_{0}, c)]

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

[MinusNegated →^{tex} “MinusNegated”]

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

eqReflexivity

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

[eqReflexivity →^{stmt} SystemQ ⊢ ∀\underline{x}: \underline{x} = \underline{x}]

[eqReflexivity →^{tex} “eqReflexivity”]

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

eqSymmetry

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

[eqSymmetry →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} = \underline{y} ⊢ \underline{y} = \underline{x}]

[eqSymmetry →^{tex} “eqSymmetry”]

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

eqTransitivity

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

[eqTransitivity →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ \underline{y} = \underline{z} ⊢ \underline{x} = \underline{z}]

[eqTransitivity →^{tex} “eqTransitivity”]

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

eqTransitivity4

[eqTransitivity4 →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: \underline{x} = \underline{y} ⊢ \underline{y} = \underline{z} ⊢ \underline{z} = \underline{u} ⊢ eqTransitivity ⊳ \underline{x} = \underline{y} ⊳ \underline{y} = \underline{z} ≫ \underline{x} = \underline{z}; eqTransitivity ⊳ \underline{x} = \underline{z} ⊳ \underline{z} = \underline{u} ≫ \underline{x} = \underline{u}⌉, p_{0}, c)]

[eqTransitivity4 →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: \underline{x} = \underline{y} ⊢ \underline{y} = \underline{z} ⊢ \underline{z} = \underline{u} ⊢ \underline{x} = \underline{u}]

[eqTransitivity4 →^{tex} “eqTransitivity4”]

[eqTransitivity4 →^{pyk} “lemma eqTransitivity4”]

eqTransitivity5

[eqTransitivity5 →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: ∀\underline{v}: \underline{x} = \underline{y} ⊢ \underline{y} = \underline{z} ⊢ \underline{z} = \underline{u} ⊢ \underline{u} = \underline{v} ⊢ eqTransitivity4 ⊳ \underline{x} = \underline{y} ⊳ \underline{y} = \underline{z} ⊳ \underline{z} = \underline{u} ≫ \underline{x} = \underline{u}; eqTransitivity ⊳ \underline{x} = \underline{u} ⊳ \underline{u} = \underline{v} ≫ \underline{x} = \underline{v}⌉, p_{0}, c)]

[eqTransitivity5 →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: ∀\underline{v}: \underline{x} = \underline{y} ⊢ \underline{y} = \underline{z} ⊢ \underline{z} = \underline{u} ⊢ \underline{u} = \underline{v} ⊢ \underline{x} = \underline{v}]

[eqTransitivity5 →^{tex} “eqTransitivity5”]

[eqTransitivity5 →^{pyk} “lemma eqTransitivity5”]

eqTransitivity6

[eqTransitivity6 →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: ∀\underline{v}: ∀\underline{w}: \underline{x} = \underline{y} ⊢ \underline{y} = \underline{z} ⊢ \underline{z} = \underline{u} ⊢ \underline{u} = \underline{v} ⊢ \underline{v} = \underline{w} ⊢ eqTransitivity5 ⊳ \underline{x} = \underline{y} ⊳ \underline{y} = \underline{z} ⊳ \underline{z} = \underline{u} ⊳ \underline{u} = \underline{v} ≫ \underline{x} = \underline{v}; eqTransitivity ⊳ \underline{x} = \underline{v} ⊳ \underline{v} = \underline{w} ≫ \underline{x} = \underline{w}⌉, p_{0}, c)]

[eqTransitivity6 →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: ∀\underline{v}: ∀\underline{w}: \underline{x} = \underline{y} ⊢ \underline{y} = \underline{z} ⊢ \underline{z} = \underline{u} ⊢ \underline{u} = \underline{v} ⊢ \underline{v} = \underline{w} ⊢ \underline{x} = \underline{w}] [eqTransitivity6 →^{tex} “eqTransitivity6”]

[eqTransitivity6 →^{pyk} “lemma eqTransitivity6”]

AddEquations

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

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

[AddEquations →^{tex} “AddEquations”]

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

SubtractEquations

[SubtractEquations →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: (\underline{x} + \underline{z}) = (\underline{y} + \underline{u}) ⊢ \underline{z} = \underline{u} ⊢ eqAddition ⊳ (\underline{x} + \underline{z}) = (\underline{y} + \underline{u}) ≫ ((\underline{x} + \underline{z}) + (−u\underline{z})) = ((\underline{y} + \underline{u}) + (−u\underline{z})); plus0Left ≫ (0 + \underline{z}) = \underline{z}; eqTransitivity ⊳ (0 + \underline{z}) = \underline{z} ⊳ \underline{z} = \underline{u} ≫ (0 + \underline{z}) = \underline{u}; PositiveToRight(Eq) ⊳ (0 + \underline{z}) = \underline{u} ≫ 0 = (\underline{u} + (−u\underline{z})); eqSymmetry ⊳ 0 = (\underline{u} + (−u\underline{z})) ≫ (\underline{u} + (−u\underline{z})) = 0; EqAdditionLeft ⊳ (\underline{u} + (−u\underline{z})) = 0 ≫ (\underline{y} + (\underline{u} + (−u\underline{z}))) = (\underline{y}+0); plusAssociativity ≫ ((\underline{y}+\underline{u})+(−u\underline{z})) = (\underline{y}+(\underline{u}+(−u\underline{z}))); plus0 ≫ (\underline{y}+0) = \underline{y}; eqTransitivity4 ⊳ ((\underline{y} + \underline{u}) + (−u\underline{z})) = (\underline{y} + (\underline{u} + (−u\underline{z}))) ⊳ (\underline{y} + (\underline{u} + (−u\underline{z}))) = (\underline{y} + 0) ⊳ (\underline{y} + 0) = \underline{y} ≫ ((\underline{y} + \underline{u}) + (−u\underline{z})) = \underline{y}; x = x + y − y ≫ \underline{x} = ((\underline{x} + \underline{z}) + (−u\underline{z})); eqTransitivity4 ⊳ \underline{x} = ((\underline{x} + \underline{z}) + (−u\underline{z})) ⊳ ((\underline{x} + \underline{z}) + (−u\underline{z})) = ((\underline{y} + \underline{u}) + (−u\underline{z})) ⊳ ((\underline{y} + \underline{u}) + (−u\underline{z})) = \underline{y} ≫ \underline{x} = \underline{y}⌉, p_{0}, c)]

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

[SubtractEquations →^{tex} “SubtractEquations”]

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

SubtractEquationsLeft

[SubtractEquationsLeft →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: (\underline{x} + \underline{z}) = (\underline{y} + \underline{u}) ⊢ \underline{x} = \underline{y} ⊢ plusCommutativity ≫ (\underline{z} + \underline{x}) = (\underline{x} + \underline{z}); plusCommutativity ≫ (\underline{y} + \underline{u}) = (\underline{u} + \underline{y}); eqTransitivity4 ⊳ (\underline{z} + \underline{x}) = (\underline{x} + \underline{z}) ⊳ (\underline{x} + \underline{z}) = (\underline{y} + \underline{u}) ⊳ (\underline{y} + \underline{u}) = (\underline{u} + \underline{y}) ≫ (\underline{z} + \underline{x}) = (\underline{u} + \underline{y}); SubtractEquations ⊳ (\underline{z} + \underline{x}) = (\underline{u} + \underline{y}) ⊳ \underline{x} = \underline{y} ≫ \underline{z} = \underline{u}⌉, p_{0}, c)]

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

[SubtractEquationsLeft →^{tex} “SubtractEquationsLeft”]

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

MultiplyEquations

[MultiplyEquations →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: \underline{x} = \underline{y} ⊢ \underline{z} = \underline{u} ⊢ eqMultiplication ⊳ \underline{x} = \underline{y} ≫ (\underline{x} ∗ \underline{z}) = (\underline{y} ∗ \underline{z}); EqMultiplicationLeft ⊳ \underline{z} = \underline{u} ≫ (\underline{y} ∗ \underline{z}) = (\underline{y} ∗ \underline{u}); eqTransitivity ⊳ (\underline{x} ∗ \underline{z}) = (\underline{y} ∗ \underline{z}) ⊳ (\underline{y} ∗ \underline{z}) = (\underline{y} ∗ \underline{u}) ≫ (\underline{x} ∗ \underline{z}) = (\underline{y} ∗ \underline{u})⌉, p_{0}, c)]

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

[MultiplyEquations →^{tex} “MultiplyEquations”]

[MultiplyEquations →^{pyk} “lemma multiplyEquations”]

EqNegated

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

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

[EqNegated →^{tex} “EqNegated”]

[EqNegated →^{pyk} “lemma eqNegated”]

PositiveToRight(Eq)

[PositiveToRight(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} + \underline{y}) + (−u\underline{y})) = (\underline{z} + (−u\underline{y})); x = x + y − y ≫ \underline{x} = ((\underline{x} + \underline{y}) + (−u\underline{y})); eqTransitivity ⊳ \underline{x} = ((\underline{x} + \underline{y}) + (−u\underline{y})) ⊳ ((\underline{x} + \underline{y}) + (−u\underline{y})) = (\underline{z} + (−u\underline{y})) ≫ \underline{x} = (\underline{z} + (−u\underline{y}))⌉, p_{0}, c)] [PositiveToRight(Eq) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: (\underline{x} + \underline{y}) = \underline{z} ⊢ \underline{x} = (\underline{z} + (−u\underline{y}))]

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

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

PositiveToLeft(Eq)(1term)

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

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

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

[PositiveToLeft(Eq)(1term) →^{pyk} “lemma positiveToLeft(Eq)(1 term)”]

NegativeToLeft(Eq)

[NegativeToLeft(Eq) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = (\underline{y} + (−u\underline{z})) ⊢ eqAddition⊳\underline{x} = (\underline{y}+(−u\underline{z})) ≫ (\underline{x}+\underline{z}) = ((\underline{y}+(−u\underline{z}))+\underline{z}); Three2threeTerms ≫ ((\underline{y} + (−u\underline{z})) + \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}; eqTransitivity4 ⊳ (\underline{x} + \underline{z}) = ((\underline{y} + (−u\underline{z})) + \underline{z}) ⊳ ((\underline{y} + (−u\underline{z})) + \underline{z}) = ((\underline{y} + \underline{z}) + (−u\underline{z})) ⊳ ((\underline{y} + \underline{z}) + (−u\underline{z})) = \underline{y} ≫ (\underline{x} + \underline{z}) = \underline{y}⌉, p_{0}, c)]

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

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

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

NonreciprocalToRight(Eq)(1term)

[NonreciprocalToRight(Eq)(1term) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: (\underline{x} ∗ \underline{y}) = 1 ⊢ eqMultiplication⊳(\underline{x}∗\underline{y}) = 1 ≫ ((\underline{x}∗\underline{y})∗rec\underline{y}) = (1∗rec\underline{y}); 0 < 1 ≫ ˙¬ (0 <= 1 ⇒ ˙¬ ( ˙¬ (0 = 1)n)n)n; PositiveNonzero ⊳ ˙¬ (0 <= 1 ⇒ ˙¬ ( ˙¬ (0 = 1)n)n)n ≫ ˙¬ (1 = 0)n; eqSymmetry ⊳ (\underline{x} ∗ \underline{y}) = 1 ≫ 1 = (\underline{x} ∗ \underline{y}); SubNeqLeft ⊳ 1 = (\underline{x} ∗ \underline{y}) ⊳ ˙¬ (1 = 0)n ≫ ˙¬ ((\underline{x} ∗ \underline{y}) = 0)n; NonzeroProduct(2) ⊳ ˙¬ ((\underline{x} ∗ \underline{y}) = 0)n ≫ ˙¬ (\underline{y} = 0)n; x = x ∗ y ∗ (1/y) ⊳ ˙¬ (\underline{y} = 0)n ≫ \underline{x} = ((\underline{x} ∗ \underline{y}) ∗ rec\underline{y}); times1Left ≫ (1 ∗ rec\underline{y}) = rec\underline{y}; eqTransitivity4 ⊳ \underline{x} = ((\underline{x} ∗ \underline{y}) ∗ rec\underline{y}) ⊳ ((\underline{x} ∗ \underline{y}) ∗ rec\underline{y}) = (1 ∗ rec\underline{y}) ⊳ (1 ∗ rec\underline{y}) = rec\underline{y} ≫ \underline{x} = rec\underline{y}⌉, p_{0}, c)]

[NonreciprocalToRight(Eq)(1term) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: (\underline{x}∗\underline{y}) = 1 ⊢ \underline{x} = rec\underline{y}]

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

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

PlusAssociativity(4terms)

[PlusAssociativity(4terms) →^{tex} “PlusAssociativity(4 terms)”]

[PlusAssociativity(4terms) →^{pyk} “lemma plusAssociativity(4 terms)”]

LessNeq

[LessNeq →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⊢ Repetition ⊳ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ≫ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n; SecondConjunct⊳ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ≫ ˙¬ (\underline{x} = \underline{y})n⌉, p_{0}, c)]

[LessNeq →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⊢ ˙¬ (\underline{x} = \underline{y})n]

[LessNeq →^{tex} “LessNeq”]

[LessNeq →^{pyk} “lemma lessNeq”]

NeqSymmetry

[NeqSymmetry →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: \underline{y} = \underline{x} ⊢ eqSymmetry ⊳ \underline{y} = \underline{x} ≫ \underline{x} = \underline{y}; ∀\underline{x}: ∀\underline{y}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: \underline{y} = \underline{x} ⊢ \underline{x} = \underline{y} ≫ \underline{y} = \underline{x} ⇒ \underline{x} = \underline{y}; ˙¬ (\underline{x} = \underline{y})n ⊢ MT ⊳ \underline{y} = \underline{x} ⇒ \underline{x} = \underline{y} ⊳ ˙¬ (\underline{x} = \underline{y})n ≫ ˙¬ (\underline{y} = \underline{x})n⌉, p_{0}, c)]

[NeqSymmetry →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ (\underline{x} = \underline{y})n ⊢ ˙¬ (\underline{y} = \underline{x})n]

[NeqSymmetry →^{tex} “NeqSymmetry”]

[NeqSymmetry →^{pyk} “lemma neqSymmetry”]

NeqNegated

[NeqNegated →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ (\underline{x} = \underline{y})n ⊢ (−u\underline{x}) = (−u\underline{y}) ⊢ EqNegated ⊳ (−u\underline{x}) = (−u\underline{y}) ≫ (−u(−u\underline{x})) = (−u(−u\underline{y})); DoubleMinus ≫ (−u(−u\underline{x})) = \underline{x}; eqSymmetry ⊳ (−u(−u\underline{x})) = \underline{x} ≫ \underline{x} = (−u(−u\underline{x})); DoubleMinus ≫ (−u(−u\underline{y})) = \underline{y}; eqTransitivity4 ⊳ \underline{x} = (−u(−u\underline{x})) ⊳ (−u(−u\underline{x})) = (−u(−u\underline{y})) ⊳ (−u(−u\underline{y})) = \underline{y} ≫ \underline{x} = \underline{y}; FromContradiction ⊳ \underline{x} = \underline{y} ⊳ ˙¬ (\underline{x} = \underline{y})n ≫ ˙¬ ((−u\underline{x}) = (−u\underline{y}))n; ∀\underline{x}: ∀\underline{y}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: ˙¬ (\underline{x} = \underline{y})n ⊢ (−u\underline{x}) = (−u\underline{y}) ⊢ ˙¬ ((−u\underline{x}) = (−u\underline{y}))n ≫ ˙¬ (\underline{x} = \underline{y})n ⇒ (−u\underline{x}) = (−u\underline{y}) ⇒ ˙¬ ((−u\underline{x}) = (−u\underline{y}))n; ˙¬ (\underline{x} = \underline{y})n ⊢ MP ⊳ ˙¬ (\underline{x} = \underline{y})n ⇒ (−u\underline{x}) = (−u\underline{y}) ⇒ ˙¬ ((−u\underline{x}) = (−u\underline{y}))n ⊳ ˙¬ (\underline{x} = \underline{y})n ≫ (−u\underline{x}) = (−u\underline{y}) ⇒ ˙¬ ((−u\underline{x}) = (−u\underline{y}))n; prop lemma imply negation ⊳ (−u\underline{x}) = (−u\underline{y}) ⇒ ˙¬ ((−u\underline{x}) = (−u\underline{y}))n ≫ ˙¬ ((−u\underline{x}) = (−u\underline{y}))n⌉, p_{0}, c)]

[NeqNegated →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ (\underline{x} = \underline{y})n ⊢ ˙¬ ((−u\underline{x}) = (−u\underline{y}))n]

[NeqNegated →^{tex} “NeqNegated”]

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

SubNeqRight

[SubNeqRight →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ ˙¬ (\underline{z} = \underline{x})n ⊢ NeqSymmetry ⊳ ˙¬ (\underline{z} = \underline{x})n ≫ ˙¬ (\underline{x} = \underline{z})n; SubNeqLeft ⊳ \underline{x} = \underline{y} ⊳ ˙¬ (\underline{x} = \underline{z})n ≫ ˙¬ (\underline{y} = \underline{z})n; NeqSymmetry ⊳ ˙¬ (\underline{y} = \underline{z})n ≫ ˙¬ (\underline{z} = \underline{y})n⌉, p_{0}, c)]

[SubNeqRight →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ ˙¬ (\underline{z} = \underline{x})n ⊢ ˙¬ (\underline{z} = \underline{y})n]

[SubNeqRight →^{tex} “SubNeqRight”]

[SubNeqRight →^{pyk} “lemma subNeqRight”]

SubNeqLeft

[SubNeqLeft →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ ˙¬ (\underline{x} = \underline{z})n ⊢ EqualityAxiom ≫ \underline{y} = \underline{x} ⇒ \underline{y} = \underline{z} ⇒ \underline{x} = \underline{z}; eqSymmetry ⊳ \underline{x} = \underline{y} ≫ \underline{y} = \underline{x}; MP ⊳ \underline{y} = \underline{x} ⇒ \underline{y} = \underline{z} ⇒ \underline{x} = \underline{z} ⊳ \underline{y} = \underline{x} ≫ \underline{y} = \underline{z} ⇒ \underline{x} = \underline{z}; Contrapositive ⊳ \underline{y} = \underline{z} ⇒ \underline{x} = \underline{z} ≫ ˙¬ (\underline{x} = \underline{z})n ⇒ ˙¬ (\underline{y} = \underline{z})n; MP ⊳ ˙¬ (\underline{x} = \underline{z})n ⇒ ˙¬ (\underline{y} = \underline{z})n ⊳ ˙¬ (\underline{x} = \underline{z})n ≫ ˙¬ (\underline{y} = \underline{z})n⌉, p_{0}, c)]

[SubNeqLeft →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ ˙¬ (\underline{x} = \underline{z})n ⊢ ˙¬ (\underline{y} = \underline{z})n]

[SubNeqLeft →^{tex} “SubNeqLeft”]

[SubNeqLeft →^{pyk} “lemma subNeqLeft”]

NegativeToRight(Neq)(1term)

[NegativeToRight(Neq)(1term) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} = \underline{y} ⊢ PositiveToLeft(Eq)(1term) ⊳ \underline{x} = \underline{y} ≫ (\underline{x} + (−u\underline{y})) = 0; ∀\underline{x}: ∀\underline{y}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: \underline{x} = \underline{y} ⊢ (\underline{x} + (−u\underline{y})) = 0 ≫ \underline{x} = \underline{y} ⇒ (\underline{x} + (−u\underline{y})) = 0; ˙¬ ((\underline{x} + (−u\underline{y})) = 0)n ⊢ MT ⊳ \underline{x} = \underline{y} ⇒ (\underline{x} + (−u\underline{y})) = 0 ⊳ ˙¬ ((\underline{x} + (−u\underline{y})) = 0)n ≫ ˙¬ (\underline{x} = \underline{y})n⌉, p_{0}, c)]

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

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

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

NeqAddition

[NeqAddition →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ (\underline{x} = \underline{y})n ⊢ (\underline{x} + \underline{z}) = (\underline{y} + \underline{z}) ⊢ eqReflexivity ≫ \underline{z} = \underline{z}; SubtractEquations ⊳ (\underline{x} + \underline{z}) = (\underline{y} + \underline{z}) ⊳ \underline{z} = \underline{z} ≫ \underline{x} = \underline{y}; FromContradiction ⊳ \underline{x} = \underline{y} ⊳ ˙¬ (\underline{x} = \underline{y})n ≫ ˙¬ ((\underline{x} + \underline{z}) = (\underline{y} + \underline{z}))n; ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ (\underline{x} = \underline{y})n ⊢ (\underline{x} + \underline{z}) = (\underline{y} + \underline{z}) ⊢ ˙¬ ((\underline{x} + \underline{z}) = (\underline{y} + \underline{z}))n ≫ ˙¬ (\underline{x} = \underline{y})n ⇒ (\underline{x} + \underline{z}) = (\underline{y} + \underline{z}) ⇒ ˙¬ ((\underline{x} + \underline{z}) = (\underline{y} + \underline{z}))n; ˙¬ (\underline{x} = \underline{y})n ⊢ MP ⊳ ˙¬ (\underline{x} = \underline{y})n ⇒ (\underline{x} + \underline{z}) = (\underline{y} + \underline{z}) ⇒ ˙¬ ((\underline{x} + \underline{z}) = (\underline{y} + \underline{z}))n ⊳ ˙¬ (\underline{x} = \underline{y})n ≫ (\underline{x}+\underline{z}) = (\underline{y} +\underline{z}) ⇒ ˙¬ ((\underline{x}+\underline{z}) = (\underline{y} +\underline{z}))n; prop lemma imply negation⊳(\underline{x}+\underline{z}) = (\underline{y} + \underline{z}) ⇒ ˙¬ ((\underline{x} + \underline{z}) = (\underline{y} + \underline{z}))n ≫ ˙¬ ((\underline{x} + \underline{z}) = (\underline{y} + \underline{z}))n⌉, p_{0}, c)]

[NeqAddition →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ (\underline{x} = \underline{y})n ⊢ ˙¬ ((\underline{x} + \underline{z}) = (\underline{y} + \underline{z}))n]

[NeqAddition →^{tex} “NeqAddition”]

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

NeqMultiplication

[NeqMultiplication →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ (\underline{z} = 0)n ⊢ ˙¬ (\underline{x} = \underline{y})n ⊢ (\underline{x} ∗ \underline{z}) = (\underline{y} ∗ \underline{z}) ⊢ x = x ∗ y ∗ (1/y) ⊳ ˙¬ (\underline{z} = 0)n ≫ \underline{x} = ((\underline{x} ∗ \underline{z}) ∗ rec\underline{z}); eqMultiplication ⊳ (\underline{x} ∗ \underline{z}) = (\underline{y} ∗ \underline{z}) ≫ ((\underline{x} ∗ \underline{z}) ∗ rec\underline{z}) = ((\underline{y} ∗ \underline{z}) ∗ rec\underline{z}); x = x∗y∗(1/y)⊳ ˙¬ (\underline{z} = 0)n ≫ \underline{y} = ((\underline{y}∗\underline{z})∗rec\underline{z}); eqSymmetry⊳\underline{y} = ((\underline{y}∗\underline{z})∗rec\underline{z}) ≫ ((\underline{y} ∗ \underline{z}) ∗ rec\underline{z}) = \underline{y}; eqTransitivity4 ⊳ \underline{x} = ((\underline{x} ∗ \underline{z}) ∗ rec\underline{z}) ⊳ ((\underline{x} ∗ \underline{z}) ∗ rec\underline{z}) = ((\underline{y} ∗ \underline{z}) ∗ rec\underline{z}) ⊳ ((\underline{y} ∗ \underline{z}) ∗ rec\underline{z}) = \underline{y} ≫ \underline{x} = \underline{y}; FromContradiction ⊳ \underline{x} = \underline{y} ⊳ ˙¬ (\underline{x} = \underline{y})n ≫ ˙¬ ((\underline{x} ∗ \underline{z}) = (\underline{y} ∗ \underline{z}))n; ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ (\underline{z} = 0)n ⊢ ˙¬ (\underline{x} = \underline{y})n ⊢ (\underline{x} ∗ \underline{z}) = (\underline{y} ∗ \underline{z}) ⊢ ˙¬ ((\underline{x} ∗ \underline{z}) = (\underline{y} ∗ \underline{z}))n ≫ ˙¬ (\underline{z} = 0)n ⇒ ˙¬ (\underline{x} = \underline{y})n ⇒ (\underline{x} ∗ \underline{z}) = (\underline{y} ∗ \underline{z}) ⇒ ˙¬ ((\underline{x} ∗ \underline{z}) = (\underline{y} ∗ \underline{z}))n; ˙¬ (\underline{z} = 0)n ⊢ ˙¬ (\underline{x} = \underline{y})n ⊢ MP2 ⊳ ˙¬ (\underline{z} = 0)n ⇒ ˙¬ (\underline{x} = \underline{y})n ⇒ (\underline{x} ∗ \underline{z}) = (\underline{y} ∗ \underline{z}) ⇒ ˙¬ ((\underline{x} ∗ \underline{z}) = (\underline{y} ∗ \underline{z}))n ⊳ ˙¬ (\underline{z} = 0)n ⊳ ˙¬ (\underline{x} = \underline{y})n ≫ (\underline{x} ∗ \underline{z}) = (\underline{y} ∗ \underline{z}) ⇒ ˙¬ ((\underline{x} ∗ \underline{z}) = (\underline{y} ∗ \underline{z}))n; prop lemma imply negation ⊳ (\underline{x} ∗ \underline{z}) = (\underline{y} ∗ \underline{z}) ⇒ ˙¬ ((\underline{x} ∗ \underline{z}) = (\underline{y} ∗ \underline{z}))n ≫ ˙¬ ((\underline{x} ∗ \underline{z}) = (\underline{y} ∗ \underline{z}))n⌉, p_{0}, c)]

[NeqMultiplication →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ (\underline{z} = 0)n ⊢ ˙¬ (\underline{x} = \underline{y})n ⊢ ˙¬ ((\underline{x} ∗ \underline{z}) = (\underline{y} ∗ \underline{z}))n]

[NeqMultiplication →^{tex} “NeqMultiplication”]

[NeqMultiplication →^{pyk} “lemma neqMultiplication”]

NonzeroProduct(2)

[NonzeroProduct(2) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: \underline{y} = 0 ⊢ EqMultiplicationLeft ⊳ \underline{y} = 0 ≫ (\underline{x} ∗ \underline{y}) = (\underline{x} ∗ 0); x ∗ 0 = 0 ≫ (\underline{x} ∗ 0) = 0; eqTransitivity ⊳ (\underline{x} ∗ \underline{y}) = (\underline{x} ∗ 0) ⊳ (\underline{x} ∗ 0) = 0 ≫ (\underline{x} ∗ \underline{y}) = 0; ∀\underline{x}: ∀\underline{y}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: \underline{y} = 0 ⊢ (\underline{x} ∗ \underline{y}) = 0 ≫ \underline{y} = 0 ⇒ (\underline{x} ∗ \underline{y}) = 0; ˙¬ ((\underline{x} ∗ \underline{y}) = 0)n ⊢ MT ⊳ \underline{y} = 0 ⇒ (\underline{x} ∗ \underline{y}) = 0 ⊳ ˙¬ ((\underline{x} ∗ \underline{y}) = 0)n ≫ ˙¬ (\underline{y} = 0)n⌉, p_{0}, c)]

[NonzeroProduct(2) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ ((\underline{x} ∗ \underline{y}) = 0)n ⊢ ˙¬ (\underline{y} = 0)n]

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

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

UStelescope(+1)

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

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

TelescopeBound(Base)

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

[TelescopeBound(Base) →^{pyk} “lemma telescopeBound base”]

TelescopeBound(Indu)

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

[TelescopeBound(Indu) →^{pyk} “lemma telescopeBound indu”]

TelescopeBound

[TelescopeBound →^{tex} “TelescopeBound”] [TelescopeBound →^{pyk} “lemma telescopeBound”]

IntervalSize(Base)

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

IntervalSize(Indu)

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

IntervalSize

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

XS < US

[XS < US →^{tex} “XS<US”] [XS < US →^{pyk} “lemma XSlessUS”]

lemma USdecreasing(+1)

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

CloseUS

[CloseUS →^{tex} “CloseUS”] [CloseUS →^{pyk} “lemma closeUS”]

CloseUS(n + 1)

[CloseUS(n + 1) →^{tex} “CloseUS(n+1)”]

[CloseUS(n + 1) →^{pyk} “lemma closeUS(n+1)”]

AllNegated(Imply)

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

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

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

[AllNegated(Imply) →^{pyk} “pred lemma allNegated(Imply)”]

ExistNegated(Imply)

[ExistNegated(Imply) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{a}: ˙¬ ( ˙¬ (∀_{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; RemoveDoubleNeg ⊳ ˙¬ ( ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{a})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)]

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

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

[ExistNegated(Imply) →^{pyk} “pred lemma existNegated(Imply)”]

IntroExist(Helper)

[IntroExist(Helper) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{(v1)}: ∀\underline{a}: ∀\underline{b}: ⟨ ˙¬ (\underline{a})n≡ ˙¬ (\underline{b})n|\underline{(v1)}:==\underline{x}⟩_{Me} ⊢⊢ ∀_{obj}\underline{(v1)}: ˙¬ (\underline{b})n ⊢ A4 @ \underline{x} ⊳ ⊳ ⟨ ˙¬ (\underline{a})n≡ ˙¬ (\underline{b})n|\underline{(v1)}:==\underline{x}⟩_{Me}⊳∀_{obj}\underline{(v1)}: ˙¬ (\underline{b})n ≫ ˙¬ (\underline{a})n; ∀\underline{x}: ∀\underline{(v1)}: ∀\underline{a}: ∀\underline{b}: Ded⊳ ∀\underline{x}: ∀\underline{(v1)}: ∀\underline{a}: ∀\underline{b}: ⟨ ˙¬ (\underline{a})n≡ ˙¬ (\underline{b})n|\underline{(v1)}:==\underline{x}⟩_{Me} ⊢⊢ ∀_{obj}\underline{(v1)}: ˙¬ (\underline{b})n ⊢ ˙¬ (\underline{a})n ≫ ⟨ ˙¬ (\underline{a})n≡ ˙¬ (\underline{b})n|\underline{(v1)}:==\underline{x}⟩_{Me} ⊢⊢ ∀_{obj}\underline{(v1)}: ˙¬ (\underline{b})n ⇒ ˙¬ (\underline{a})n⌉, p_{0}, c)]

[IntroExist(Helper) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{(v1)}: ∀\underline{a}: ∀\underline{b}: ⟨ ˙¬ (\underline{a})n≡ ˙¬ (\underline{b})n|\underline{(v1)}:==\underline{x}⟩_{Me} ⊢⊢ ∀_{obj}\underline{(v1)}: ˙¬ (\underline{b})n ⇒ ˙¬ (\underline{a})n]

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

[IntroExist(Helper) →^{pyk} “pred lemma intro exist helper”]

IntroExist

[IntroExist →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{(v1)}: ∀\underline{a}: ∀\underline{b}: ⟨ ˙¬ (\underline{a})n≡ ˙¬ (\underline{b})n|\underline{(v1)}:==\underline{x}⟩_{Me} ⊢⊢ IntroExist(Helper) @ \underline{x} ⊳ ⊳ ⟨ ˙¬ (\underline{a})n≡ ˙¬ (\underline{b})n|\underline{(v1)}:==\underline{x}⟩_{Me} ≫ ∀_{obj}\underline{(v1)}: ˙¬ (\underline{b})n ⇒ ˙¬ (\underline{a})n; \underline{a} ⊢ AddDoubleNeg ⊳ \underline{a} ≫ ˙¬ ( ˙¬ (\underline{a})n)n; MT ⊳ ∀_{obj}\underline{(v1)}: ˙¬ (\underline{b})n ⇒ ˙¬ (\underline{a})n ⊳ ˙¬ ( ˙¬ (\underline{a})n)n ≫ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{b})n)n; Repetition ⊳ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{b})n)n ≫ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{b})n)n⌉, p_{0}, c)]

[IntroExist →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{(v1)}: ∀\underline{a}: ∀\underline{b}: ⟨ ˙¬ (\underline{a})n≡ ˙¬ (\underline{b})n|\underline{(v1)}:==\underline{x}⟩_{Me} ⊢⊢ \underline{a} ⊢ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{b})n)n]

[IntroExist →^{tex} “IntroExist”]

[IntroExist →^{pyk} “pred lemma intro exist”]

ExistMP

[ExistMP →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{a})n)n ⊢ ˙¬ (\underline{b})n ⊢ MT ⊳ \underline{a} ⇒ \underline{b} ⊳ ˙¬ (\underline{b})n ≫ ˙¬ (\underline{a})n; Gen ⊳ ˙¬ (\underline{a})n ≫ ∀_{obj}\underline{(v1)}: ˙¬ (\underline{a})n; Repetition ⊳ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{a})n)n ≫ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{a})n)n; FromContradiction ⊳ ∀_{obj}\underline{(v1)}: ˙¬ (\underline{a})n ⊳ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{a})n)n ≫ ˙¬ ( ˙¬ (\underline{b})n)n; ∀\underline{(v1)}: ∀\underline{a}: ∀\underline{b}: Ded ⊳ ∀\underline{(v1)}: ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{a})n)n ⊢ ˙¬ (\underline{b})n ⊢ ˙¬ ( ˙¬ (\underline{b})n)n ≫ \underline{a} ⇒ \underline{b} ⇒ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{a})n)n ⇒ ˙¬ (\underline{b})n ⇒ ˙¬ ( ˙¬ (\underline{b})n)n; \underline{a} ⇒ \underline{b} ⊢ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{a})n)n ⊢ MP2 ⊳ \underline{a} ⇒ \underline{b} ⇒ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{a})n)n ⇒ ˙¬ (\underline{b})n ⇒ ˙¬ ( ˙¬ (\underline{b})n)n ⊳ \underline{a} ⇒ \underline{b} ⊳ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{a})n)n ≫ ˙¬ (\underline{b})n ⇒ ˙¬ ( ˙¬ (\underline{b})n)n; prop lemma imply negation ⊳ ˙¬ (\underline{b})n ⇒ ˙¬ ( ˙¬ (\underline{b})n)n ≫ ˙¬ ( ˙¬ (\underline{b})n)n; RemoveDoubleNeg ⊳ ˙¬ ( ˙¬ (\underline{b})n)n ≫ \underline{b}⌉, p_{0}, c)] [ExistMP →^{stmt} SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{a})n)n ⊢ \underline{b}]

[ExistMP →^{tex} “ExistMP”]

[ExistMP →^{pyk} “pred lemma exist mp”]

ExistMP2

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

[ExistMP2 →^{stmt} SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⊢ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{a})n)n ⊢ ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (\underline{b})n)n ⊢ \underline{c}]

[ExistMP2 →^{tex} “ExistMP2”]

[ExistMP2 →^{pyk} “pred lemma exist mp2”]

TwiceExistMP

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

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

[TwiceExistMP →^{tex} “TwiceExistMP”]

[TwiceExistMP →^{pyk} “pred lemma 2exist mp”]

TwiceExistMP2

[TwiceExistMP2 →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{(v3)}: ∀\underline{(v4)}: ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⊢ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ ( ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (\underline{a})n)n)n)n ⊢ ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ ( ˙¬ (∀_{obj}\underline{(v4)}: ˙¬ (\underline{b})n)n)n)n ⊢ TwiceExistMP ⊳ \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⊳ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ ( ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (\underline{a})n)n)n)n ≫ \underline{b} ⇒ \underline{c}; TwiceExistMP ⊳ \underline{b} ⇒ \underline{c} ⊳ ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ ( ˙¬ (∀_{obj}\underline{(v4)}: ˙¬ (\underline{b})n)n)n)n ≫ \underline{c}⌉, p_{0}, c)]

[TwiceExistMP2 →^{stmt} SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{(v3)}: ∀\underline{(v4)}: ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⊢ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ ( ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (\underline{a})n)n)n)n ⊢ ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ ( ˙¬ (∀_{obj}\underline{(v4)}: ˙¬ (\underline{b})n)n)n)n ⊢ \underline{c}]

[TwiceExistMP2 →^{tex} “TwiceExistMP2”]

[TwiceExistMP2 →^{pyk} “pred lemma 2exist mp2”]

EAE − MP

[EAE − MP →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(v2)}: ∀\underline{(v3)}: ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ ∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ (\underline{a})n)n ⊢ A4 @ \underline{(v2)} ⊳ ∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ (\underline{a})n)n ≫ ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ (\underline{a})n)n; ExistMP ⊳ \underline{a} ⇒ \underline{b} ⊳ ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ (\underline{a})n)n ≫ \underline{b}; ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{(v3)}: ∀\underline{a}: ∀\underline{b}: Ded ⊳ ∀\underline{(v2)}: ∀\underline{(v3)}: ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ ∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ (\underline{a})n)n ⊢ \underline{b} ≫ \underline{a} ⇒ \underline{b} ⇒ ∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ (\underline{a})n)n ⇒ \underline{b}; \underline{a} ⇒ \underline{b} ⊢ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ (\underline{a})n)n)n)n ⊢ MP ⊳ \underline{a} ⇒ \underline{b} ⇒ ∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ (\underline{a})n)n ⇒ \underline{b} ⊳ \underline{a} ⇒ \underline{b} ≫ ∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ (\underline{a})n)n ⇒ \underline{b}; ExistMP ⊳ ∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ (\underline{a})n)n ⇒ \underline{b} ⊳ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ (\underline{a})n)n)n)n ≫ \underline{b}⌉, p_{0}, c)]

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

[EAE − MP →^{tex} “EAE-MP”]

[EAE − MP →^{pyk} “pred lemma EAE mp”]

AddAll

[AddAll →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ ∀_{obj}\underline{(v1)}: \underline{a} ⊢ A4 ⊳ ∀_{obj}\underline{(v1)}: \underline{a} ≫ \underline{a}; MP ⊳ \underline{a} ⇒ \underline{b} ⊳ \underline{a} ≫ \underline{b}; Gen ⊳ \underline{b} ≫ ∀_{obj}\underline{(v1)}: \underline{b}; ∀\underline{(v1)}: ∀\underline{a}: ∀\underline{b}: Ded ⊳ ∀\underline{(v1)}: ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ ∀_{obj}\underline{(v1)}: \underline{a} ⊢ ∀_{obj}\underline{(v1)}: \underline{b} ≫ \underline{a} ⇒ \underline{b} ⇒ ∀_{obj}\underline{(v1)}: \underline{a} ⇒ ∀_{obj}\underline{(v1)}: \underline{b}; \underline{a} ⇒ \underline{b} ⊢ MP ⊳ \underline{a} ⇒ \underline{b} ⇒ ∀_{obj}\underline{(v1)}: \underline{a} ⇒ ∀_{obj}\underline{(v1)}: \underline{b} ⊳ \underline{a} ⇒ \underline{b} ≫ ∀_{obj}\underline{(v1)}: \underline{a} ⇒ ∀_{obj}\underline{(v1)}: \underline{b}⌉, p_{0}, c)]

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

[AddAll →^{tex} “AddAll ”]

[AddAll →^{pyk} “pred lemma addAll”]

AddExist(Helper1)

[AddExist(Helper1) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{y}: ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ⟨ ˙¬ (\underline{b})n≡ ˙¬ (\underline{d})n|\underline{(v2)}:==\underline{y}⟩_{Me} ⊢⊢ \underline{a} ⇒ \underline{b} ⊢ \underline{c} ⇒ \underline{a} ⊢ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{c})n)n ⊢ ∀_{obj}\underline{(v2)}: ˙¬ (\underline{d})n ⊢ A4 @ \underline{y} ⊳ ∀_{obj}\underline{(v2)}: ˙¬ (\underline{d})n ≫ ˙¬ (\underline{b})n; MT ⊳ \underline{a} ⇒ \underline{b} ⊳ ˙¬ (\underline{b})n ≫ ˙¬ (\underline{a})n; MT ⊳ \underline{c} ⇒ \underline{a} ⊳ ˙¬ (\underline{a})n ≫ ˙¬ (\underline{c})n; Gen ⊳ ˙¬ (\underline{c})n ≫ ∀_{obj}\underline{(v1)}: ˙¬ (\underline{c})n; Repetition ⊳ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{c})n)n ≫ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{c})n)n; FromContradiction ⊳ ∀_{obj}\underline{(v1)}: ˙¬ (\underline{c})n ⊳ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{c})n)n ≫ ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (\underline{d})n)n; ∀\underline{y}: ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: Ded ⊳ ∀\underline{y}: ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ⟨ ˙¬ (\underline{b})n≡ ˙¬ (\underline{d})n|\underline{(v2)}:==\underline{y}⟩_{Me} ⊢⊢ \underline{a} ⇒ \underline{b} ⊢ \underline{c} ⇒ \underline{a} ⊢ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{c})n)n ⊢ ∀_{obj}\underline{(v2)}: ˙¬ (\underline{d})n ⊢ ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (\underline{d})n)n ≫ ⟨ ˙¬ (\underline{b})n≡ ˙¬ (\underline{d})n|\underline{(v2)}:==\underline{y}⟩_{Me} ⊢⊢ \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{a} ⇒ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{c})n)n ⇒ ∀_{obj}\underline{(v2)}: ˙¬ (\underline{d})n ⇒ ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (\underline{d})n)n⌉, p_{0}, c)]

[AddExist(Helper1) →^{stmt} SystemQ ⊢ ∀\underline{y}: ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ⟨ ˙¬ (\underline{b})n≡ ˙¬ (\underline{d})n|\underline{(v2)}:==\underline{y}⟩_{Me} ⊢⊢ \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{a} ⇒ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{c})n)n ⇒ ∀_{obj}\underline{(v2)}: ˙¬ (\underline{d})n ⇒ ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (\underline{d})n)n]

[AddExist(Helper1) →^{tex} “AddExist(Helper1)”]

[AddExist(Helper1) →^{pyk} “pred lemma addExist helper1”]

AddExist(Helper2)

[AddExist(Helper2) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{y}: ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ⟨ ˙¬ (\underline{b})n≡ ˙¬ (\underline{d})n|\underline{(v2)}:==\underline{y}⟩_{Me} ⊢⊢ \underline{a} ⇒ \underline{b} ⊢ \underline{c} ⇒ \underline{a} ⊢ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{c})n)n ⊢ AddExist(Helper1) ⊳ ⊳ ⟨ ˙¬ (\underline{b})n≡ ˙¬ (\underline{d})n|\underline{(v2)}:==\underline{y}⟩_{Me} ≫ \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{a} ⇒ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{c})n)n ⇒ ∀_{obj}\underline{(v2)}: ˙¬ (\underline{d})n ⇒ ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (\underline{d})n)n; MP3 ⊳ \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{a} ⇒ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{c})n)n ⇒ ∀_{obj}\underline{(v2)}: ˙¬ (\underline{d})n ⇒ ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (\underline{d})n)n ⊳ \underline{a} ⇒ \underline{b} ⊳ \underline{c} ⇒ \underline{a} ⊳ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{c})n)n ≫ ∀_{obj}\underline{(v2)}: ˙¬ (\underline{d})n ⇒ ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (\underline{d})n)n; prop lemma imply negation ⊳ ∀_{obj}\underline{(v2)}: ˙¬ (\underline{d})n ⇒ ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (\underline{d})n)n ≫ ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (\underline{d})n)n; Repetition ⊳ ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (\underline{d})n)n ≫ ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (\underline{d})n)n; ∀\underline{y}: ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: Ded ⊳ ∀\underline{y}: ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ⟨ ˙¬ (\underline{b})n≡ ˙¬ (\underline{d})n|\underline{(v2)}:==\underline{y}⟩_{Me} ⊢⊢ \underline{a} ⇒ \underline{b} ⊢ \underline{c} ⇒ \underline{a} ⊢ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{c})n)n ⊢ ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (\underline{d})n)n ≫ ⟨ ˙¬ (\underline{b})n≡ ˙¬ (\underline{d})n|\underline{(v2)}:==\underline{y}⟩_{Me} ⊢⊢ \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{a} ⇒ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{c})n)n ⇒ ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (\underline{d})n)n⌉, p_{0}, c)]

[AddExist(Helper2) →^{stmt} SystemQ ⊢ ∀\underline{y}: ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ⟨ ˙¬ (\underline{b})n≡ ˙¬ (\underline{d})n|\underline{(v2)}:==\underline{y}⟩_{Me} ⊢⊢ \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{a} ⇒ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{c})n)n ⇒ ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (\underline{d})n)n]

[AddExist(Helper2) →^{tex} “AddExist(Helper2)”] [AddExist(Helper2) →^{pyk} “pred lemma addExist helper2”]

AddExist

[AddExist →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{y}: ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ⟨ ˙¬ (\underline{b})n≡ ˙¬ (\underline{d})n|\underline{(v2)}:==\underline{y}⟩_{Me} ⊢⊢ \underline{a} ⇒ \underline{b} ⊢ \underline{c} ⇒ \underline{a} ⊢ AddExist(Helper2) ⊳ ⊳ ⟨ ˙¬ (\underline{b})n≡ ˙¬ (\underline{d})n|\underline{(v2)}:==\underline{y}⟩_{Me} ≫ \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{a} ⇒ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{c})n)n ⇒ ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (\underline{d})n)n; MP2 ⊳ \underline{a} ⇒ \underline{b} ⇒ \underline{c} ⇒ \underline{a} ⇒ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{c})n)n ⇒ ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (\underline{d})n)n ⊳ \underline{a} ⇒ \underline{b} ⊳ \underline{c} ⇒ \underline{a} ≫ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{c})n)n ⇒ ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (\underline{d})n)n⌉, p_{0}, c)]

[AddExist →^{stmt} SystemQ ⊢ ∀\underline{y}: ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ∀\underline{d}: ⟨ ˙¬ (\underline{b})n≡ ˙¬ (\underline{d})n|\underline{(v2)}:==\underline{y}⟩_{Me} ⊢⊢ \underline{a} ⇒ \underline{b} ⊢ \underline{c} ⇒ \underline{a} ⊢ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{c})n)n ⇒ ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (\underline{d})n)n]

[AddExist →^{tex} “AddExist”]

[AddExist →^{pyk} “pred lemma addExist”]

AddExist(SimpleAnt)

[AddExist(SimpleAnt) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{y}: ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{a}: ∀\underline{b}: ∀\underline{d}: ⟨ ˙¬ (\underline{b})n≡ ˙¬ (\underline{d})n|\underline{(v2)}:==\underline{y}⟩_{Me} ⊢⊢ \underline{a} ⇒ \underline{b} ⊢ AutoImply ≫ \underline{a} ⇒ \underline{a}; AddExist @ \underline{y} ⊳ ⊳ ⟨ ˙¬ (\underline{b})n≡ ˙¬ (\underline{d})n|\underline{(v2)}:==\underline{y}⟩_{Me} ⊳ \underline{a} ⇒ \underline{b} ⊳ \underline{a} ⇒ \underline{a} ≫ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{a})n)n ⇒ ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (\underline{d})n)n⌉, p_{0}, c)]

[AddExist(SimpleAnt) →^{stmt} SystemQ ⊢ ∀\underline{y}: ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{a}: ∀\underline{b}: ∀\underline{d}: ⟨ ˙¬ (\underline{b})n≡ ˙¬ (\underline{d})n|\underline{(v2)}:==\underline{y}⟩_{Me} ⊢⊢ \underline{a} ⇒ \underline{b} ⊢ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{a})n)n ⇒ ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (\underline{d})n)n]

[AddExist(SimpleAnt) →^{tex} “AddExist(SimpleAnt)”]

[AddExist(SimpleAnt) →^{pyk} “pred lemma addExist(SimpleAnt)”]

AddExist(Simple)

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

[AddExist(Simple) →^{stmt} SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{a}: ∀\underline{b}: \underline{a} ⇒ \underline{b} ⊢ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (\underline{a})n)n ⇒ ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (\underline{b})n)n] [AddExist(Simple) →^{tex} “AddExist(Simple)”]

[AddExist(Simple) →^{pyk} “pred lemma addExist(Simple)”]

AddEAE

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

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

[AddEAE →^{tex} “AddEAE”]

[AddEAE →^{pyk} “pred lemma addEAE”]

AEA − negated

[AEA − negated →^{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 ⊢ AllNegated(Imply) ≫ ˙¬ (∀_{obj}\underline{(v3)}: \underline{a})n ⇒ ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ ( ˙¬ (\underline{a})n)n)n; AddAll ⊳ ˙¬ (∀_{obj}\underline{(v3)}: \underline{a})n ⇒ ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ ( ˙¬ (\underline{a})n)n)n ≫ ∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: \underline{a})n ⇒ ∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ ( ˙¬ (\underline{a})n)n)n; ExistNegated(Imply) ≫ ˙¬ ( ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: \underline{a})n)n)n ⇒ ∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: \underline{a})n; ImplyTransitivity ⊳ ˙¬ ( ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: \underline{a})n)n)n ⇒ ∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: \underline{a})n ⊳ ∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: \underline{a})n ⇒ ∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ ( ˙¬ (\underline{a})n)n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: \underline{a})n)n)n ⇒ ∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ ( ˙¬ (\underline{a})n)n)n; AddExist(Simple) ⊳ ˙¬ ( ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: \underline{a})n)n)n ⇒ ∀_{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)n)n; AllNegated(Imply) ≫ ˙¬ (∀_{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; ImplyTransitivity ⊳ ˙¬ (∀_{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)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; MP ⊳ ˙¬ (∀_{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 ≫ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ ( ˙¬ (\underline{a})n)n)n)n)n⌉, p_{0}, c)]

[AEA − negated →^{stmt} SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{(v3)}: ∀\underline{a}: ˙¬ (∀_{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]

[AEA − negated →^{tex} “AEA-negated”]

[AEA − negated →^{pyk} “pred lemma AEAnegated”]

EEA − negated

[EEA − negated →^{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 ⊢ AllNegated(Imply) ≫ ˙¬ (∀_{obj}\underline{(v3)}: \underline{a})n ⇒ ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ ( ˙¬ (\underline{a})n)n)n; AddAll ⊳ ˙¬ (∀_{obj}\underline{(v3)}: \underline{a})n ⇒ ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ ( ˙¬ (\underline{a})n)n)n ≫ ∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: \underline{a})n ⇒ ∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ ( ˙¬ (\underline{a})n)n)n; ExistNegated(Imply) ≫ ˙¬ ( ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: \underline{a})n)n)n ⇒ ∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: \underline{a})n; ImplyTransitivity ⊳ ˙¬ ( ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: \underline{a})n)n)n ⇒ ∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: \underline{a})n ⊳ ∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: \underline{a})n ⇒ ∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ ( ˙¬ (\underline{a})n)n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: \underline{a})n)n)n ⇒ ∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ ( ˙¬ (\underline{a})n)n)n; AddAll ⊳ ˙¬ ( ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: \underline{a})n)n)n ⇒ ∀_{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 ⇒ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (∀_{obj}\underline{(v3)}: ˙¬ ( ˙¬ (\underline{a})n)n)n; ExistNegated(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; 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 ⊳ ∀_{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 ≫ ˙¬ ( ˙¬ (∀_{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)] [EEA − negated →^{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]

[EEA − negated →^{tex} “EEA-negated”]

[EEA − negated →^{pyk} “pred lemma EEAnegated”]

Induction

[Induction →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ⟨\underline{b}≡\underline{a}|\underline{(v1)}:==0⟩_{Me} ⊢⊢ ⟨\underline{c}≡\underline{a}|\underline{(v1)}:==(\underline{(v1)} + 1)⟩_{Me} ⊢⊢ \underline{b} ⊢ \underline{a} ⇒ \underline{c} ⊢ Gen ⊳ \underline{a} ⇒ \underline{c} ≫ ∀_{obj}\underline{(v1)}: \underline{a} ⇒ \underline{c}; InductionAxiom ⊳ ⊳ ⟨\underline{b}≡\underline{a}|\underline{(v1)}:==0⟩_{Me} ⊳ ⊳ ⟨\underline{c}≡\underline{a}|\underline{(v1)}:==(\underline{(v1)} + 1)⟩_{Me} ≫ \underline{b} ⇒ ∀_{obj}\underline{(v1)}: \underline{a} ⇒ \underline{c} ⇒ ∀_{obj}\underline{(v1)}: \underline{a}; MP2 ⊳ \underline{b} ⇒ ∀_{obj}\underline{(v1)}: \underline{a} ⇒ \underline{c} ⇒ ∀_{obj}\underline{(v1)}: \underline{a}⊳\underline{b}⊳∀_{obj}\underline{(v1)}: \underline{a} ⇒ \underline{c} ≫ ∀_{obj}\underline{(v1)}: \underline{a}; A4 @ \underline{(v1)}⊳∀_{obj}\underline{(v1)}: \underline{a} ≫ \underline{a}⌉, p_{0}, c)]

[Induction →^{stmt} SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{a}: ∀\underline{b}: ∀\underline{c}: ⟨\underline{b}≡\underline{a}|\underline{(v1)}:==0⟩_{Me} ⊢⊢ ⟨\underline{c}≡\underline{a}|\underline{(v1)}:==(\underline{(v1)} + 1)⟩_{Me} ⊢⊢ \underline{b} ⊢ \underline{a} ⇒ \underline{c} ⊢ \underline{a}]

[Induction →^{tex} “Induction”]

[Induction →^{pyk} “lemma induction”]

leqAntisymmetry

[leqAntisymmetry →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ \underline{y} <= \underline{x} ⊢ leqAntisymmetryAxiom ≫ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x} ⇒ \underline{x} = \underline{y}; MP2 ⊳ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x} ⇒ \underline{x} = \underline{y} ⊳ \underline{x} <= \underline{y} ⊳ \underline{y} <= \underline{x} ≫ \underline{x} = \underline{y}⌉, p_{0}, c)]

[leqAntisymmetry →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ \underline{y} <= \underline{x} ⊢ \underline{x} = \underline{y}]

[leqAntisymmetry →^{tex} “leqAntisymmetry”]

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

leqTransitivity

[leqTransitivity →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= \underline{y} ⊢ \underline{y} <= \underline{z} ⊢ leqTransitivityAxiom ≫ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{z} ⇒ \underline{x} <= \underline{z}; MP2 ⊳ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{z} ⇒ \underline{x} <= \underline{z} ⊳ \underline{x} <= \underline{y} ⊳ \underline{y} <= \underline{z} ≫ \underline{x} <= \underline{z}⌉, p_{0}, c)]

[leqTransitivity →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= \underline{y} ⊢ \underline{y} <= \underline{z} ⊢ \underline{x} <= \underline{z}]

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

leqAddition

[leqAddition →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= \underline{y} ⊢ leqAdditionAxiom ≫ \underline{x} <= \underline{y} ⇒ (\underline{x} + \underline{z}) <= (\underline{y} + \underline{z}); MP ⊳ \underline{x} <= \underline{y} ⇒ (\underline{x} + \underline{z}) <= (\underline{y} + \underline{z}) ⊳ \underline{x} <= \underline{y} ≫ (\underline{x} + \underline{z}) <= (\underline{y} + \underline{z})⌉, p_{0}, c)]

[leqAddition →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= \underline{y} ⊢ (\underline{x} + \underline{z}) <= (\underline{y} + \underline{z})]

[leqAddition →^{tex} “leqAddition”]

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

leqMultiplication

[leqMultiplication →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: 0 <= \underline{z} ⊢ \underline{x} <= \underline{y} ⊢ leqMultiplicationAxiom ≫ 0 <= \underline{z} ⇒ \underline{x} <= \underline{y} ⇒ (\underline{x} ∗ \underline{z}) <= (\underline{y} ∗ \underline{z}); MP2 ⊳ 0 <= \underline{z} ⇒ \underline{x} <= \underline{y} ⇒ (\underline{x} ∗ \underline{z}) <= (\underline{y} ∗ \underline{z}) ⊳ 0 <= \underline{z} ⊳ \underline{x} <= \underline{y} ≫ (\underline{x} ∗ \underline{z}) <= (\underline{y} ∗ \underline{z})⌉, p_{0}, c)]

[leqMultiplication →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: 0 <= \underline{z} ⊢ \underline{x} <= \underline{y} ⊢ (\underline{x}∗\underline{z}) <= (\underline{y}∗\underline{z})]

[leqMultiplication →^{tex} “leqMultiplication”]

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

Reciprocal

[Reciprocal →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ˙¬ (\underline{x} = 0)n ⊢ ReciprocalAxiom ≫ ˙¬ (\underline{x} = 0)n ⇒ (\underline{x} ∗ rec\underline{x}) = 1; MP ⊳ ˙¬ (\underline{x} = 0)n ⇒ (\underline{x} ∗ rec\underline{x}) = 1 ⊳ ˙¬ (\underline{x} = 0)n ≫ (\underline{x} ∗ rec\underline{x}) = 1⌉, p_{0}, c)]

[Reciprocal →^{stmt} SystemQ ⊢ ∀\underline{x}: ˙¬ (\underline{x} = 0)n ⊢ (\underline{x} ∗ rec\underline{x}) = 1]

[Reciprocal →^{tex} “Reciprocal”]

[Reciprocal →^{pyk} “lemma reciprocal”]

Equality

[Equality →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ \underline{x} = \underline{z} ⊢ EqualityAxiom ≫ \underline{x} = \underline{y} ⇒ \underline{x} = \underline{z} ⇒ \underline{y} = \underline{z}; MP2 ⊳ \underline{x} = \underline{y} ⇒ \underline{x} = \underline{z} ⇒ \underline{y} = \underline{z} ⊳ \underline{x} = \underline{y} ⊳ \underline{x} = \underline{z} ≫ \underline{y} = \underline{z}⌉, p_{0}, c)] [Equality →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ \underline{x} = \underline{z} ⊢ \underline{y} = \underline{z}] [Equality →^{tex} “Equality”] [Equality →^{pyk} “lemma equality”]

eqLeq

[eqLeq →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} = \underline{y} ⊢ EqLeqAxiom ≫ \underline{x} = \underline{y} ⇒ \underline{x} <= \underline{y}; MP ⊳ \underline{x} = \underline{y} ⇒ \underline{x} <= \underline{y} ⊳ \underline{x} = \underline{y} ≫ \underline{x} <= \underline{y}⌉, p_{0}, c)] [eqLeq →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} = \underline{y} ⊢ \underline{x} <= \underline{y}] [eqLeq →^{tex} “eqLeq”] [eqLeq →^{pyk} “lemma eqLeq”]

eqAddition

[eqAddition →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ EqAdditionAxiom ≫ \underline{x} = \underline{y} ⇒ (\underline{x} + \underline{z}) = (\underline{y} + \underline{z}); MP ⊳ \underline{x} = \underline{y} ⇒ (\underline{x} + \underline{z}) = (\underline{y} + \underline{z}) ⊳ \underline{x} = \underline{y} ≫ (\underline{x} + \underline{z}) = (\underline{y} + \underline{z})⌉, p_{0}, c)] [eqAddition →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ (\underline{x} + \underline{z}) = (\underline{y} + \underline{z})] [eqAddition →^{tex} “eqAddition”] [eqAddition →^{pyk} “lemma eqAddition”]

eqMultiplication

[eqMultiplication →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ EqMultiplicationAxiom ≫ \underline{x} = \underline{y} ⇒ (\underline{x} ∗ \underline{z}) = (\underline{y} ∗ \underline{z}); MP ⊳ \underline{x} = \underline{y} ⇒ (\underline{x} ∗ \underline{z}) = (\underline{y} ∗ \underline{z}) ⊳ \underline{x} = \underline{y} ≫ (\underline{x} ∗ \underline{z}) = (\underline{y} ∗ \underline{z})⌉, p_{0}, c)] [eqMultiplication →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ (\underline{x} ∗ \underline{z}) = (\underline{y} ∗ \underline{z})] [eqMultiplication →^{tex} “eqMultiplication”] [eqMultiplication →^{pyk} “lemma eqMultiplication”]

LeqMultiplicationLeft

[LeqMultiplicationLeft →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: 0 <= \underline{z} ⊢ \underline{x} <= \underline{y} ⊢ leqMultiplication⊳0 <= \underline{z}⊳\underline{x} <= \underline{y} ≫ (\underline{x}∗\underline{z}) <= (\underline{y}∗\underline{z}); timesCommutativity ≫ (\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{z} ∗ \underline{x}) <= (\underline{y} ∗ \underline{z}); timesCommutativity ≫ (\underline{y} ∗ \underline{z}) = (\underline{z} ∗ \underline{y}); subLeqRight ⊳ (\underline{y} ∗ \underline{z}) = (\underline{z} ∗ \underline{y}) ⊳ (\underline{z} ∗ \underline{x}) <= (\underline{y} ∗ \underline{z}) ≫ (\underline{z} ∗ \underline{x}) <= (\underline{z} ∗ \underline{y})⌉, p_{0}, c)]

[LeqMultiplicationLeft →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: 0 <= \underline{z} ⊢ \underline{x} <= \underline{y} ⊢ (\underline{z} ∗ \underline{x}) <= (\underline{z} ∗ \underline{y})]

[LeqMultiplicationLeft →^{tex} “LeqMultiplicationLeft ”]

[LeqMultiplicationLeft →^{pyk} “lemma leqMultiplicationLeft”]

LeqLessEq

[LeqLessEq →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ ˙¬ ( ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n)n ⊢ fromNotLess ⊳ ˙¬ ( ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n)n ≫ \underline{y} <= \underline{x}; leqAntisymmetry ⊳ \underline{x} <= \underline{y} ⊳ \underline{y} <= \underline{x} ≫ \underline{x} = \underline{y}; ∀\underline{x}: ∀\underline{y}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ ˙¬ ( ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n)n ⊢ \underline{x} = \underline{y} ≫ \underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n)n ⇒ \underline{x} = \underline{y}; \underline{x} <= \underline{y} ⊢ MP ⊳ \underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n)n ⇒ \underline{x} = \underline{y} ⊳ \underline{x} <= \underline{y} ≫ ˙¬ ( ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n)n ⇒ \underline{x} = \underline{y}; Repetition ⊳ ˙¬ ( ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n)n ⇒ \underline{x} = \underline{y} ≫ ˙¬ ( ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n)n ⇒ \underline{x} = \underline{y}⌉, p_{0}, c)]

[LeqLessEq →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ ˙¬ ( ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n)n ⇒ \underline{x} = \underline{y}]

[LeqLessEq →^{tex} “LeqLessEq”]

[LeqLessEq →^{pyk} “lemma leqLessEq”]

LessLeq

[LessLeq →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⊢ Repetition ⊳ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ≫ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n; FirstConjunct ⊳ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ≫ \underline{x} <= \underline{y}⌉, p_{0}, c)]

[LessLeq →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⊢ \underline{x} <= \underline{y}]

[LessLeq →^{tex} “LessLeq”]

[LessLeq →^{pyk} “lemma lessLeq”]

FromLeqGeq

[FromLeqGeq →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⇒ \underline{a} ⊢ \underline{y} <= \underline{x} ⇒ \underline{a} ⊢ leqTotality ≫ ˙¬ (\underline{x} <= \underline{y})n ⇒ \underline{y} <= \underline{x}; FromDisjuncts ⊳ ˙¬ (\underline{x} <= \underline{y})n ⇒ \underline{y} <= \underline{x} ⊳ \underline{x} <= \underline{y} ⇒ \underline{a} ⊳ \underline{y} <= \underline{x} ⇒ \underline{a} ≫ \underline{a}⌉, p_{0}, c)]

[FromLeqGeq →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⇒ \underline{a} ⊢ \underline{y} <= \underline{x} ⇒ \underline{a} ⊢ \underline{a}]

[FromLeqGeq →^{tex} “FromLeqGeq”]

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

subLeqRight

[subLeqRight →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ \underline{z} <= \underline{x} ⊢ eqLeq ⊳ \underline{x} = \underline{y} ≫ \underline{x} <= \underline{y}; leqTransitivity ⊳ \underline{z} <= \underline{x} ⊳ \underline{x} <= \underline{y} ≫ \underline{z} <= \underline{y}⌉, p_{0}, c)]

[subLeqRight →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ \underline{z} <= \underline{x} ⊢ \underline{z} <= \underline{y}]

[subLeqRight →^{tex} “subLeqRight”]

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

subLeqLeft

[subLeqLeft →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ \underline{x} <= \underline{z} ⊢ eqSymmetry ⊳ \underline{x} = \underline{y} ≫ \underline{y} = \underline{x}; eqLeq ⊳ \underline{y} = \underline{x} ≫ \underline{y} <= \underline{x}; leqTransitivity ⊳ \underline{y} <= \underline{x} ⊳ \underline{x} <= \underline{z} ≫ \underline{y} <= \underline{z}⌉, p_{0}, c)]

[subLeqLeft →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ \underline{x} <= \underline{z} ⊢ \underline{y} <= \underline{z}]

[subLeqLeft →^{tex} “subLeqLeft”]

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

Leq + 1

[Leq + 1 →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ 0 < 1 ≫ ˙¬ (0 <= 1 ⇒ ˙¬ ( ˙¬ (0 = 1)n)n)n; LessAdditionLeft ⊳ ˙¬ (0 <= 1 ⇒ ˙¬ ( ˙¬ (0 = 1)n)n)n ≫ ˙¬ ((\underline{y} + 0) <= (\underline{y} + 1) ⇒ ˙¬ ( ˙¬ ((\underline{y} + 0) = (\underline{y} + 1))n)n)n; plus0 ≫ (\underline{y} + 0) = \underline{y}; SubLessLeft ⊳ (\underline{y} + 0) = \underline{y} ⊳ ˙¬ ((\underline{y} + 0) <= (\underline{y} + 1) ⇒ ˙¬ ( ˙¬ ((\underline{y} + 0) = (\underline{y} + 1))n)n)n ≫ ˙¬ (\underline{y} <= (\underline{y} + 1) ⇒ ˙¬ ( ˙¬ (\underline{y} = (\underline{y} + 1))n)n)n; leqLessTransitivity ⊳ \underline{x} <= \underline{y} ⊳ ˙¬ (\underline{y} <= (\underline{y} + 1) ⇒ ˙¬ ( ˙¬ (\underline{y} = (\underline{y} + 1))n)n)n ≫ ˙¬ (\underline{x} <= (\underline{y} + 1) ⇒ ˙¬ ( ˙¬ (\underline{x} = (\underline{y} + 1))n)n)n⌉, p_{0}, c)]

[Leq + 1 →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ ˙¬ (\underline{x} <= (\underline{y} + 1) ⇒ ˙¬ ( ˙¬ (\underline{x} = (\underline{y} + 1))n)n)n]

[Leq + 1 →^{tex} “Leq+1”]

[Leq + 1 →^{pyk} “lemma leqPlus1”]

PositiveToRight(Leq)

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

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

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

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

PositiveToRight(Leq)(1term)

[PositiveToRight(Leq)(1term) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{y}: ∀\underline{z}: \underline{y} <= \underline{z} ⊢ plus0Left ≫ (0 + \underline{y}) = \underline{y}; eqSymmetry ⊳ (0 + \underline{y}) = \underline{y} ≫ \underline{y} = (0 + \underline{y}); subLeqLeft ⊳ \underline{y} = (0 + \underline{y}) ⊳ \underline{y} <= \underline{z} ≫ (0 + \underline{y}) <= \underline{z}; PositiveToRight(Leq) ⊳ (0 + \underline{y}) <= \underline{z} ≫ 0 <= (\underline{z} + (−u\underline{y}))⌉, p_{0}, c)]

[PositiveToRight(Leq)(1term) →^{stmt} SystemQ ⊢ ∀\underline{y}: ∀\underline{z}: \underline{y} <= \underline{z} ⊢ 0 <= (\underline{z} + (−u\underline{y}))]

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

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

lemma negativeToRight(Leq)

[lemma negativeToRight(Leq) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: (\underline{x} + (−u\underline{y})) <= \underline{z} ⊢ leqAddition ⊳ (\underline{x} + (−u\underline{y})) <= \underline{z} ≫ ((\underline{x} + (−u\underline{y})) + \underline{y}) <= (\underline{z} + \underline{y}); x = x + y − y ≫ \underline{x} = ((\underline{x} + \underline{y}) + (−u\underline{y})); Three2threeTerms ≫ ((\underline{x} + \underline{y}) + (−u\underline{y})) = ((\underline{x} + (−u\underline{y})) + \underline{y}); eqTransitivity ⊳ \underline{x} = ((\underline{x} + \underline{y}) + (−u\underline{y})) ⊳ ((\underline{x} + \underline{y}) + (−u\underline{y})) = ((\underline{x} + (−u\underline{y})) + \underline{y}) ≫ \underline{x} = ((\underline{x} + (−u\underline{y})) + \underline{y}); eqSymmetry ⊳ \underline{x} = ((\underline{x} + (−u\underline{y})) + \underline{y}) ≫ ((\underline{x} + (−u\underline{y})) + \underline{y}) = \underline{x}; subLeqLeft ⊳ ((\underline{x} + (−u\underline{y})) + \underline{y}) = \underline{x} ⊳ ((\underline{x} + (−u\underline{y})) + \underline{y}) <= (\underline{z} + \underline{y}) ≫ \underline{x} <= (\underline{z} + \underline{y})⌉, p_{0}, c)]

[lemma negativeToRight(Leq) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: (\underline{x} + (−u\underline{y})) <= \underline{z} ⊢ \underline{x} <= (\underline{z} + \underline{y})]

[lemma negativeToRight(Leq) →^{pyk} “lemma negativeToRight(Leq)”]

PositiveToLeft(Leq)

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

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

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

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

negativeToLeft(Leq)

[negativeToLeft(Leq) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= (\underline{y} + (−u\underline{z})) ⊢ leqAddition ⊳ \underline{x} <= (\underline{y} + (−u\underline{z})) ≫ (\underline{x} + \underline{z}) <= ((\underline{y} + (−u\underline{z})) + \underline{z}); x = x + y − y ≫ \underline{y} = ((\underline{y} + \underline{z}) + (−u\underline{z})); Three2threeTerms ≫ ((\underline{y} + \underline{z}) + (−u\underline{z})) = ((\underline{y} + (−u\underline{z})) + \underline{z}); eqTransitivity ⊳ \underline{y} = ((\underline{y} + \underline{z}) + (−u\underline{z})) ⊳ ((\underline{y} + \underline{z}) + (−u\underline{z})) = ((\underline{y} + (−u\underline{z})) + \underline{z}) ≫ \underline{y} = ((\underline{y} + (−u\underline{z})) + \underline{z}); eqSymmetry ⊳ \underline{y} = ((\underline{y} + (−u\underline{z})) + \underline{z}) ≫ ((\underline{y} + (−u\underline{z})) + \underline{z}) = \underline{y}; subLeqRight ⊳ ((\underline{y} + (−u\underline{z})) + \underline{z}) = \underline{y} ⊳ (\underline{x} + \underline{z}) <= ((\underline{y} + (−u\underline{z})) + \underline{z}) ≫ (\underline{x} + \underline{z}) <= \underline{y}⌉, p_{0}, c)]

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

[negativeToLeft(Leq) →^{tex} “negativeToLeft(Leq)”]

[negativeToLeft(Leq) →^{pyk} “lemma negativeToLeft(Leq)”]

negativeToLeft(Leq)(1term)

[negativeToLeft(Leq)(1term) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{y}: ∀\underline{z}: 0 <= (\underline{y} + (−u\underline{z})) ⊢ negativeToLeft(Leq) ⊳ 0 <= (\underline{y} + (−u\underline{z})) ≫ (0 + \underline{z}) <= \underline{y}; plus0Left ≫ (0 + \underline{z}) = \underline{z}; subLeqLeft ⊳ (0 + \underline{z}) = \underline{z} ≫ \underline{z} <= \underline{y}⌉, p_{0}, c)]

[negativeToLeft(Leq)(1term) →^{stmt} SystemQ ⊢ ∀\underline{y}: ∀\underline{z}: 0 <= (\underline{y}+(−u\underline{z})) ⊢ \underline{z} <= \underline{y}]

[negativeToLeft(Leq)(1term) →^{tex} “negativeToLeft(Leq)(1 term)”]

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

LeqAdditionLeft

[LeqAdditionLeft →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= \underline{y} ⊢ leqAddition ⊳ \underline{x} <= \underline{y} ≫ (\underline{x} + \underline{z}) <= (\underline{y} + \underline{z}); plusCommutativity ≫ (\underline{x} + \underline{z}) = (\underline{z} + \underline{x}); plusCommutativity ≫ (\underline{y} + \underline{z}) = (\underline{z} + \underline{y}); subLeqLeft ⊳ (\underline{x} + \underline{z}) = (\underline{z} + \underline{x}) ⊳ (\underline{x} + \underline{z}) <= (\underline{y} + \underline{z}) ≫ (\underline{z} + \underline{x}) <= (\underline{y} + \underline{z}); subLeqRight ⊳ (\underline{y} + \underline{z}) = (\underline{z} + \underline{y}) ⊳ (\underline{z} + \underline{x}) <= (\underline{y} + \underline{z}) ≫ (\underline{z} + \underline{x}) <= (\underline{z} + \underline{y})⌉, p_{0}, c)]

[LeqAdditionLeft →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= \underline{y} ⊢ (\underline{z} + \underline{x}) <= (\underline{z} + \underline{y})]

[LeqAdditionLeft →^{tex} “LeqAdditionLeft”]

[LeqAdditionLeft →^{pyk} “lemma leqAdditionLeft”]

leqSubtraction

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

[leqSubtraction →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: (\underline{x} + \underline{z}) <= (\underline{y} + \underline{z}) ⊢ \underline{x} <= \underline{y}]

[leqSubtraction →^{tex} “leqSubtraction”]

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

leqSubtractionLeft

[leqSubtractionLeft →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: (\underline{z} + \underline{x}) <= (\underline{z} + \underline{y}) ⊢ plusCommutativity ≫ (\underline{z} + \underline{x}) = (\underline{x} + \underline{z}); plusCommutativity ≫ (\underline{z} + \underline{y}) = (\underline{y} + \underline{z}); subLeqLeft ⊳ (\underline{z} + \underline{x}) = (\underline{x} + \underline{z}) ⊳ (\underline{z} + \underline{x}) <= (\underline{z} + \underline{y}) ≫ (\underline{x} + \underline{z}) <= (\underline{z} + \underline{y}); subLeqRight ⊳ (\underline{z} + \underline{y}) = (\underline{y} + \underline{z}) ⊳ (\underline{x} + \underline{z}) <= (\underline{z} + \underline{y}) ≫ (\underline{x} + \underline{z}) <= (\underline{y} + \underline{z}); leqSubtraction ⊳ (\underline{x} + \underline{z}) <= (\underline{y} + \underline{z}) ≫ \underline{x} <= \underline{y}⌉, p_{0}, c)]

[leqSubtractionLeft →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: (\underline{z} + \underline{x}) <= (\underline{z} + \underline{y}) ⊢ \underline{x} <= \underline{y}]

[leqSubtractionLeft →^{tex} “leqSubtractionLeft”]

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

thirdGeq

[thirdGeq →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ leqReflexivity ≫ \underline{y} <= \underline{y}; JoinConjuncts ⊳ \underline{x} <= \underline{y} ⊳ \underline{y} <= \underline{y} ≫ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ (\underline{y} <= \underline{y})n)n; ExistIntro @ c_{Ex} @ \underline{y} ⊳ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ (\underline{y} <= \underline{y})n)n ≫ ˙¬ (\underline{x} <= c_{Ex} ⇒ ˙¬ (\underline{y} <= c_{Ex})n)n; ∀\underline{x}: ∀\underline{y}: \underline{y} <= \underline{x} ⊢ leqReflexivity ≫ \underline{x} <= \underline{x}; JoinConjuncts ⊳ \underline{x} <= \underline{x} ⊳ \underline{y} <= \underline{x} ≫ ˙¬ (\underline{x} <= \underline{x} ⇒ ˙¬ (\underline{y} <= \underline{x})n)n; ExistIntro @ c_{Ex} @ \underline{x} ⊳ ˙¬ (\underline{x} <= \underline{x} ⇒ ˙¬ (\underline{y} <= \underline{x})n)n ≫ ˙¬ (\underline{x} <= c_{Ex} ⇒ ˙¬ (\underline{y} <= c_{Ex})n)n; ∀\underline{x}: ∀\underline{y}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ ˙¬ (\underline{x} <= c_{Ex} ⇒ ˙¬ (\underline{y} <= c_{Ex})n)n ≫ \underline{x} <= \underline{y} ⇒ ˙¬ (\underline{x} <= c_{Ex} ⇒ ˙¬ (\underline{y} <= c_{Ex})n)n; Ded ⊳ ∀\underline{x}: ∀\underline{y}: \underline{y} <= \underline{x} ⊢ ˙¬ (\underline{x} <= c_{Ex} ⇒ ˙¬ (\underline{y} <= c_{Ex})n)n ≫ \underline{y} <= \underline{x} ⇒ ˙¬ (\underline{x} <= c_{Ex} ⇒ ˙¬ (\underline{y} <= c_{Ex})n)n; leqTotality ≫ ˙¬ (\underline{x} <= \underline{y})n ⇒ \underline{y} <= \underline{x}; FromDisjuncts ⊳ ˙¬ (\underline{x} <= \underline{y})n ⇒ \underline{y} <= \underline{x} ⊳ \underline{x} <= \underline{y} ⇒ ˙¬ (\underline{x} <= c_{Ex} ⇒ ˙¬ (\underline{y} <= c_{Ex})n)n ⊳ \underline{y} <= \underline{x} ⇒ ˙¬ (\underline{x} <= c_{Ex} ⇒ ˙¬ (\underline{y} <= c_{Ex})n)n ≫ ˙¬ (\underline{x} <= c_{Ex} ⇒ ˙¬ (\underline{y} <= c_{Ex})n)n⌉, p_{0}, c)]

[thirdGeq →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ (\underline{x} <= c_{Ex} ⇒ ˙¬ (\underline{y} <= c_{Ex})n)n]

[thirdGeq →^{tex} “thirdGeq”]

[thirdGeq →^{pyk} “lemma thirdGeq”]

LeqNegated

[LeqNegated →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ leqAddition ⊳ \underline{x} <= \underline{y} ≫ (\underline{x} + (−u\underline{x})) <= (\underline{y} + (−u\underline{x})); Negative ≫ (\underline{x} + (−u\underline{x})) = 0; subLeqLeft ⊳ (\underline{x} + (−u\underline{x})) = 0 ⊳ (\underline{x} + (−u\underline{x})) <= (\underline{y} + (−u\underline{x})) ≫ 0 <= (\underline{y} + (−u\underline{x})); plusCommutativity ≫ (\underline{y} + (−u\underline{x})) = ((−u\underline{x}) + \underline{y}); subLeqRight ⊳ (\underline{y} + (−u\underline{x})) = ((−u\underline{x}) + \underline{y}) ⊳ 0 <= (\underline{y} + (−u\underline{x})) ≫ 0 <= ((−u\underline{x}) + \underline{y}); leqAddition ⊳ 0 <= ((−u\underline{x}) + \underline{y}) ≫ (0 + (−u\underline{y})) <= (((−u\underline{x}) + \underline{y}) + (−u\underline{y})); plus0Left ≫ (0 + (−u\underline{y})) = (−u\underline{y}); x = x + y − y ≫ (−u\underline{x}) = (((−u\underline{x})+\underline{y})+(−u\underline{y})); eqSymmetry ⊳(−u\underline{x}) = (((−u\underline{x})+\underline{y})+(−u\underline{y})) ≫ (((−u\underline{x}) + \underline{y}) + (−u\underline{y})) = (−u\underline{x}); subLeqLeft ⊳ (0 + (−u\underline{y})) = (−u\underline{y}) ⊳ (0 + (−u\underline{y})) <= (((−u\underline{x}) + \underline{y}) + (−u\underline{y})) ≫ (−u\underline{y}) <= (((−u\underline{x}) + \underline{y}) + (−u\underline{y})); subLeqRight ⊳ (((−u\underline{x}) + \underline{y}) + (−u\underline{y})) = (−u\underline{x}) ⊳ (−u\underline{y}) <= (((−u\underline{x}) + \underline{y}) + (−u\underline{y})) ≫ (−u\underline{y}) <= (−u\underline{x})⌉, p_{0}, c)]

[LeqNegated →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ (−u\underline{y}) <= (−u\underline{x})]

[LeqNegated →^{tex} “LeqNegated”]

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

AddEquations(Leq)

[AddEquations(Leq) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: \underline{x} <= \underline{y} ⊢ \underline{z} <= \underline{u} ⊢ leqAddition ⊳ \underline{x} <= \underline{y} ≫ (\underline{x} + \underline{z}) <= (\underline{y} + \underline{z}); LeqAdditionLeft ⊳ \underline{z} <= \underline{u} ≫ (\underline{y} + \underline{z}) <= (\underline{y} + \underline{u}); leqTransitivity ⊳ (\underline{x} + \underline{z}) <= (\underline{y} + \underline{z}) ⊳ (\underline{y} + \underline{z}) <= (\underline{y} + \underline{u}) ≫ (\underline{x} + \underline{z}) <= (\underline{y} + \underline{u})⌉, p_{0}, c)]

[AddEquations(Leq) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: \underline{x} <= \underline{y} ⊢ \underline{z} <= \underline{u} ⊢ (\underline{x} + \underline{z}) <= (\underline{y} + \underline{u})]

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

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

MultiplyEquations(Leq)

[MultiplyEquations(Leq) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: 0 <= \underline{x} ⊢ 0 <= \underline{z} ⊢ \underline{x} <= \underline{y} ⊢ \underline{z} <= \underline{u} ⊢ leqMultiplication ⊳ 0 <= \underline{z} ⊳ \underline{x} <= \underline{y} ≫ (\underline{x} ∗ \underline{z}) <= (\underline{y}∗\underline{z}); leqTransitivity⊳0 <= \underline{x}⊳\underline{x} <= \underline{y} ≫ 0 <= \underline{y}; LeqMultiplicationLeft⊳0 <= \underline{y} ⊳ \underline{z} <= \underline{u} ≫ (\underline{y} ∗ \underline{z}) <= (\underline{y} ∗ \underline{u}); leqTransitivity ⊳ (\underline{x} ∗ \underline{z}) <= (\underline{y} ∗ \underline{z}) ⊳ (\underline{y} ∗ \underline{z}) <= (\underline{y} ∗ \underline{u}) ≫ (\underline{x} ∗ \underline{z}) <= (\underline{y} ∗ \underline{u})⌉, p_{0}, c)]

[MultiplyEquations(Leq) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: 0 <= \underline{x} ⊢ 0 <= \underline{z} ⊢ \underline{x} <= \underline{y} ⊢ \underline{z} <= \underline{u} ⊢ (\underline{x} ∗ \underline{z}) <= (\underline{y} ∗ \underline{u})]

[MultiplyEquations(Leq) →^{tex} “MultiplyEquations(Leq)”]

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

ThirdGeqSeries

[ThirdGeqSeries →^{tex} “ThirdGeqSeries”] [ThirdGeqSeries →^{pyk} “lemma thirdGeqSeries”]

LeqNeqLess

[LeqNeqLess →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ ˙¬ (\underline{x} = \underline{y})n ⊢ JoinConjuncts ⊳ \underline{x} <= \underline{y} ⊳ ˙¬ (\underline{x} = \underline{y})n ≫ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n; Repetition ⊳ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ≫ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n⌉, p_{0}, c)]

[LeqNeqLess →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ ˙¬ (\underline{x} = \underline{y})n ⊢ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n]

[LeqNeqLess →^{tex} “LeqNeqLess”]

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

FromLess

[FromLess →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: \underline{y} <= \underline{x} ⊢ toNotLess ⊳ \underline{y} <= \underline{x} ≫ ˙¬ ( ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n)n; ∀\underline{x}: ∀\underline{y}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: \underline{y} <= \underline{x} ⊢ ˙¬ ( ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n)n ≫ \underline{y} <= \underline{x} ⇒ ˙¬ ( ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n)n; ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⊢ AddDoubleNeg ⊳ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ≫ ˙¬ ( ˙¬ ( ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n)n)n; MT ⊳ \underline{y} <= \underline{x} ⇒ ˙¬ ( ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n)n ⊳ ˙¬ ( ˙¬ ( ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n)n)n ≫ ˙¬ (\underline{y} <= \underline{x})n⌉, p_{0}, c)]

[FromLess →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⊢ ˙¬ (\underline{y} <= \underline{x})n]

[FromLess →^{tex} “FromLess”]

[FromLess →^{pyk} “lemma fromLess”]

ToLess

[ToLess →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ ( ˙¬ (\underline{y} <= \underline{x} ⇒ ˙¬ ( ˙¬ (\underline{y} = \underline{x})n)n)n)n ⊢ fromNotLess ⊳ ˙¬ ( ˙¬ (\underline{y} <= \underline{x} ⇒ ˙¬ ( ˙¬ (\underline{y} = \underline{x})n)n)n)n ≫ \underline{x} <= \underline{y}; ∀\underline{x}: ∀\underline{y}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: ˙¬ ( ˙¬ (\underline{y} <= \underline{x} ⇒ ˙¬ ( ˙¬ (\underline{y} = \underline{x})n)n)n)n ⊢ \underline{x} <= \underline{y} ≫ ˙¬ ( ˙¬ (\underline{y} <= \underline{x} ⇒ ˙¬ ( ˙¬ (\underline{y} = \underline{x})n)n)n)n ⇒ \underline{x} <= \underline{y}; ˙¬ (\underline{x} <= \underline{y})n ⊢ NegativeMT ⊳ ˙¬ ( ˙¬ (\underline{y} <= \underline{x} ⇒ ˙¬ ( ˙¬ (\underline{y} = \underline{x})n)n)n)n ⇒ \underline{x} <= \underline{y} ⊳ ˙¬ (\underline{x} <= \underline{y})n ≫ ˙¬ (\underline{y} <= \underline{x} ⇒ ˙¬ ( ˙¬ (\underline{y} = \underline{x})n)n)n⌉, p_{0}, c)]

[ToLess →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ (\underline{x} <= \underline{y})n ⊢ ˙¬ (\underline{y} <= \underline{x} ⇒ ˙¬ ( ˙¬ (\underline{y} = \underline{x})n)n)n] [ToLess →^{tex} “ToLess”]

[ToLess →^{pyk} “lemma toLess”]

fromNotLess

[fromNotLess →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ ( ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n)n ⊢ \underline{x} <= \underline{y} ⊢ Repetition ⊳ ˙¬ ( ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n)n ≫ ˙¬ ( ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n)n; RemoveDoubleNeg ⊳ ˙¬ ( ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n)n ≫ \underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n; MP ⊳ \underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n ⊳ \underline{x} <= \underline{y} ≫ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n; RemoveDoubleNeg ⊳ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n ≫ \underline{x} = \underline{y}; eqSymmetry ⊳ \underline{x} = \underline{y} ≫ \underline{y} = \underline{x}; eqLeq ⊳ \underline{y} = \underline{x} ≫ \underline{y} <= \underline{x}; ∀\underline{x}: ∀\underline{y}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: ˙¬ ( ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n)n ⊢ \underline{x} <= \underline{y} ⊢ \underline{y} <= \underline{x} ≫ ˙¬ ( ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n)n ⇒ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x}; ˙¬ ( ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n)n ⊢ MP ⊳ ˙¬ ( ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n)n ⇒ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x} ⊳ ˙¬ ( ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n)n ≫ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x}; AutoImply ≫ \underline{y} <= \underline{x} ⇒ \underline{y} <= \underline{x}; leqTotality ≫ ˙¬ (\underline{x} <= \underline{y})n ⇒ \underline{y} <= \underline{x}; FromDisjuncts ⊳ ˙¬ (\underline{x} <= \underline{y})n ⇒ \underline{y} <= \underline{x} ⊳ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x} ⊳ \underline{y} <= \underline{x} ⇒ \underline{y} <= \underline{x} ≫ \underline{y} <= \underline{x}⌉, p_{0}, c)]

[fromNotLess →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ ( ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n)n ⊢ \underline{y} <= \underline{x}]

[fromNotLess →^{tex} “fromNotLess”]

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

toNotLess

[toNotLess →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ \underline{y} <= \underline{x} ⊢ leqAntisymmetry ⊳ \underline{y} <= \underline{x} ⊳ \underline{x} <= \underline{y} ≫ \underline{y} = \underline{x}; AddDoubleNeg ⊳ \underline{y} = \underline{x} ≫ ˙¬ ( ˙¬ (\underline{y} = \underline{x})n)n; ∀\underline{x}: ∀\underline{y}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ \underline{y} <= \underline{x} ⊢ ˙¬ ( ˙¬ (\underline{y} = \underline{x})n)n ≫ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x} ⇒ ˙¬ ( ˙¬ (\underline{y} = \underline{x})n)n; \underline{x} <= \underline{y} ⊢ MP ⊳ \underline{x} <= \underline{y} ⇒ \underline{y} <= \underline{x} ⇒ ˙¬ ( ˙¬ (\underline{y} = \underline{x})n)n ⊳ \underline{x} <= \underline{y} ≫ \underline{y} <= \underline{x} ⇒ ˙¬ ( ˙¬ (\underline{y} = \underline{x})n)n; AddDoubleNeg ⊳ \underline{y} <= \underline{x} ⇒ ˙¬ ( ˙¬ (\underline{y} = \underline{x})n)n ≫ ˙¬ ( ˙¬ (\underline{y} <= \underline{x} ⇒ ˙¬ ( ˙¬ (\underline{y} = \underline{x})n)n)n)n; Repetition ⊳ ˙¬ ( ˙¬ (\underline{y} <= \underline{x} ⇒ ˙¬ ( ˙¬ (\underline{y} = \underline{x})n)n)n)n ≫ ˙¬ ( ˙¬ (\underline{y} <= \underline{x} ⇒ ˙¬ ( ˙¬ (\underline{y} = \underline{x})n)n)n)n; Repetition ⊳ ˙¬ ( ˙¬ (\underline{y} <= \underline{x} ⇒ ˙¬ ( ˙¬ (\underline{y} = \underline{x})n)n)n)n ≫ ˙¬ ( ˙¬ (\underline{y} <= \underline{x} ⇒ ˙¬ ( ˙¬ (\underline{y} = \underline{x})n)n)n)n⌉, p_{0}, c)]

[toNotLess →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= \underline{y} ⊢ ˙¬ ( ˙¬ (\underline{y} <= \underline{x} ⇒ ˙¬ ( ˙¬ (\underline{y} = \underline{x})n)n)n)n]

[toNotLess →^{tex} “toNotLess”]

[toNotLess →^{pyk} “lemma toNotLess”]

NegativeLessPositive

[NegativeLessPositive →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ⊢ FirstConjunct ⊳ ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ≫ 0 <= \underline{x}; leqAddition ⊳ 0 <= \underline{x} ≫ (0 + (−u\underline{x})) <= (\underline{x} + (−u\underline{x})); plus0Left ≫ (0 + (−u\underline{x})) = (−u\underline{x}); Negative ≫ (\underline{x} + (−u\underline{x})) = 0; subLeqLeft ⊳ (0 + (−u\underline{x})) = (−u\underline{x}) ⊳ (0 + (−u\underline{x})) <= (\underline{x} + (−u\underline{x})) ≫ (−u\underline{x}) <= (\underline{x} + (−u\underline{x})); subLeqRight ⊳ (\underline{x} + (−u\underline{x})) = 0 ⊳ (−u\underline{x}) <= (\underline{x} + (−u\underline{x})) ≫ (−u\underline{x}) <= 0; leqLessTransitivity ⊳ (−u\underline{x}) <= 0 ⊳ ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ≫ ˙¬ ((−u\underline{x}) <= \underline{x} ⇒ ˙¬ ( ˙¬ ((−u\underline{x}) = \underline{x})n)n)n⌉, p_{0}, c)]

[NegativeLessPositive →^{stmt} SystemQ ⊢ ∀\underline{x}: ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ⊢ ˙¬ ((−u\underline{x}) <= \underline{x} ⇒ ˙¬ ( ˙¬ ((−u\underline{x}) = \underline{x})n)n)n]

[NegativeLessPositive →^{tex} “NegativeLessPositive”]

[NegativeLessPositive →^{pyk} “lemma negativeLessPositive”]

leqLessTransitivity

[leqLessTransitivity →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= \underline{y} ⊢ ˙¬ (\underline{y} <= \underline{z} ⇒ ˙¬ ( ˙¬ (\underline{y} = \underline{z})n)n)n ⊢ \underline{x} = \underline{z} ⊢ FirstConjunct ⊳ ˙¬ (\underline{y} <= \underline{z} ⇒ ˙¬ ( ˙¬ (\underline{y} = \underline{z})n)n)n ≫ \underline{y} <= \underline{z}; SecondConjunct ⊳ ˙¬ (\underline{y} <= \underline{z} ⇒ ˙¬ ( ˙¬ (\underline{y} = \underline{z})n)n)n ≫ ˙¬ (\underline{y} = \underline{z})n; subLeqLeft ⊳ \underline{x} = \underline{z} ⊳ \underline{x} <= \underline{y} ≫ \underline{z} <= \underline{y}; leqAntisymmetry ⊳ \underline{y} <= \underline{z} ⊳ \underline{z} <= \underline{y} ≫ \underline{y} = \underline{z}; FromContradiction ⊳ \underline{y} = \underline{z} ⊳ ˙¬ (\underline{y} = \underline{z})n ≫ ˙¬ (\underline{x} = \underline{z})n; ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= \underline{y} ⊢ ˙¬ (\underline{y} <= \underline{z} ⇒ ˙¬ ( ˙¬ (\underline{y} = \underline{z})n)n)n ⊢ \underline{x} = \underline{z} ⊢ ˙¬ (\underline{x} = \underline{z})n ≫ \underline{x} <= \underline{y} ⇒ ˙¬ (\underline{y} <= \underline{z} ⇒ ˙¬ ( ˙¬ (\underline{y} = \underline{z})n)n)n ⇒ \underline{x} = \underline{z} ⇒ ˙¬ (\underline{x} = \underline{z})n; \underline{x} <= \underline{y} ⊢ ˙¬ (\underline{y} <= \underline{z} ⇒ ˙¬ ( ˙¬ (\underline{y} = \underline{z})n)n)n ⊢ MP2 ⊳ \underline{x} <= \underline{y} ⇒ ˙¬ (\underline{y} <= \underline{z} ⇒ ˙¬ ( ˙¬ (\underline{y} = \underline{z})n)n)n ⇒ \underline{x} = \underline{z} ⇒ ˙¬ (\underline{x} = \underline{z})n ⊳ \underline{x} <= \underline{y} ⊳ ˙¬ (\underline{y} <= \underline{z} ⇒ ˙¬ ( ˙¬ (\underline{y} = \underline{z})n)n)n ≫ \underline{x} = \underline{z} ⇒ ˙¬ (\underline{x} = \underline{z})n; prop lemma imply negation ⊳ \underline{x} = \underline{z} ⇒ ˙¬ (\underline{x} = \underline{z})n ≫ ˙¬ (\underline{x} = \underline{z})n; FirstConjunct ⊳ ˙¬ (\underline{y} <= \underline{z} ⇒ ˙¬ ( ˙¬ (\underline{y} = \underline{z})n)n)n ≫ \underline{y} <= \underline{z}; leqTransitivity ⊳ \underline{x} <= \underline{y} ⊳ \underline{y} <= \underline{z} ≫ \underline{x} <= \underline{z}; JoinConjuncts ⊳ \underline{x} <= \underline{z} ⊳ ˙¬ (\underline{x} = \underline{z})n ≫ ˙¬ (\underline{x} <= \underline{z} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{z})n)n)n⌉, p_{0}, c)]

[leqLessTransitivity →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} <= \underline{y} ⊢ ˙¬ (\underline{y} <= \underline{z} ⇒ ˙¬ ( ˙¬ (\underline{y} = \underline{z})n)n)n ⊢ ˙¬ (\underline{x} <= \underline{z} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{z})n)n)n]

[leqLessTransitivity →^{tex} “leqLessTransitivity”]

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

LessLeqTransitivity

[LessLeqTransitivity →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⊢ \underline{y} <= \underline{z} ⊢ \underline{z} = \underline{x} ⊢ FirstConjunct ⊳ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ≫ \underline{x} <= \underline{y}; SecondConjunct ⊳ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ≫ ˙¬ (\underline{x} = \underline{y})n; subLeqRight ⊳ \underline{z} = \underline{x} ⊳ \underline{y} <= \underline{z} ≫ \underline{y} <= \underline{x}; leqAntisymmetry ⊳ \underline{x} <= \underline{y} ⊳ \underline{y} <= \underline{x} ≫ \underline{x} = \underline{y}; FromContradiction ⊳ \underline{x} = \underline{y} ⊳ ˙¬ (\underline{x} = \underline{y})n ≫ ˙¬ (\underline{z} = \underline{x})n; ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⊢ \underline{y} <= \underline{z} ⊢ \underline{z} = \underline{x} ⊢ ˙¬ (\underline{z} = \underline{x})n ≫ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⇒ \underline{y} <= \underline{z} ⇒ \underline{z} = \underline{x} ⇒ ˙¬ (\underline{z} = \underline{x})n; ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⊢ \underline{y} <= \underline{z} ⊢ MP2 ⊳ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⇒ \underline{y} <= \underline{z} ⇒ \underline{z} = \underline{x} ⇒ ˙¬ (\underline{z} = \underline{x})n ⊳ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⊳ \underline{y} <= \underline{z} ≫ \underline{z} = \underline{x} ⇒ ˙¬ (\underline{z} = \underline{x})n; prop lemma imply negation ⊳ \underline{z} = \underline{x} ⇒ ˙¬ (\underline{z} = \underline{x})n ≫ ˙¬ (\underline{z} = \underline{x})n; NeqSymmetry ⊳ ˙¬ (\underline{z} = \underline{x})n ≫ ˙¬ (\underline{x} = \underline{z})n; FirstConjunct ⊳ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ≫ \underline{x} <= \underline{y}; leqTransitivity ⊳ \underline{x} <= \underline{y} ⊳ \underline{y} <= \underline{z} ≫ \underline{x} <= \underline{z}; JoinConjuncts ⊳ \underline{x} <= \underline{z} ⊳ ˙¬ (\underline{x} = \underline{z})n ≫ ˙¬ (\underline{x} <= \underline{z} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{z})n)n)n⌉, p_{0}, c)]

[LessLeqTransitivity →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⊢ \underline{y} <= \underline{z} ⊢ ˙¬ (\underline{x} <= \underline{z} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{z})n)n)n]

[LessLeqTransitivity →^{tex} “LessLeqTransitivity”]

[LessLeqTransitivity →^{pyk} “lemma lessLeqTransitivity”]

LessTransitivity

[LessTransitivity →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⊢ ˙¬ (\underline{y} <= \underline{z} ⇒ ˙¬ ( ˙¬ (\underline{y} = \underline{z})n)n)n ⊢ FirstConjunct ⊳ ˙¬ (\underline{y} <= \underline{z} ⇒ ˙¬ ( ˙¬ (\underline{y} = \underline{z})n)n)n ≫ \underline{y} <= \underline{z}; LessLeqTransitivity ⊳ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⊳ \underline{y} <= \underline{z} ≫ ˙¬ (\underline{x} <= \underline{z} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{z})n)n)n⌉, p_{0}, c)]

[LessTransitivity →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⊢ ˙¬ (\underline{y} <= \underline{z} ⇒ ˙¬ ( ˙¬ (\underline{y} = \underline{z})n)n)n ⊢ ˙¬ (\underline{x} <= \underline{z} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{z})n)n)n]

[LessTransitivity →^{tex} “LessTransitivity”]

[LessTransitivity →^{pyk} “lemma lessTransitivity”]

LessTotality

[LessTotality →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ ( ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n)n ⊢ ˙¬ (\underline{x} = \underline{y})n ⊢ fromNotLess ⊳ ˙¬ ( ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n)n ≫ \underline{y} <= \underline{x}; NeqSymmetry⊳ ˙¬ (\underline{x} = \underline{y})n ≫ ˙¬ (\underline{y} = \underline{x})n; LeqNeqLess⊳\underline{y} <= \underline{x}⊳ ˙¬ (\underline{y} = \underline{x})n ≫ ˙¬ (\underline{y} <= \underline{x} ⇒ ˙¬ ( ˙¬ (\underline{y} = \underline{x})n)n)n; ∀\underline{x}: ∀\underline{y}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: ˙¬ ( ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n)n ⊢ ˙¬ (\underline{x} = \underline{y})n ⊢ ˙¬ (\underline{y} <= \underline{x} ⇒ ˙¬ ( ˙¬ (\underline{y} = \underline{x})n)n)n ≫ ˙¬ ( ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n)n ⇒ ˙¬ (\underline{x} = \underline{y})n ⇒ ˙¬ (\underline{y} <= \underline{x} ⇒ ˙¬ ( ˙¬ (\underline{y} = \underline{x})n)n)n; Repetition ⊳ ˙¬ ( ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n)n ⇒ ˙¬ (\underline{x} = \underline{y})n ⇒ ˙¬ (\underline{y} <= \underline{x} ⇒ ˙¬ ( ˙¬ (\underline{y} = \underline{x})n)n)n ≫ ˙¬ ( ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n)n ⇒ ˙¬ (\underline{x} = \underline{y})n ⇒ ˙¬ (\underline{y} <= \underline{x} ⇒ ˙¬ ( ˙¬ (\underline{y} = \underline{x})n)n)n⌉, p_{0}, c)]

[LessTotality →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ ( ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n)n ⇒ ˙¬ (\underline{x} = \underline{y})n ⇒ ˙¬ (\underline{y} <= \underline{x} ⇒ ˙¬ ( ˙¬ (\underline{y} = \underline{x})n)n)n]

[LessTotality →^{tex} “LessTotality”]

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

SubLessRight

[SubLessRight →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ ˙¬ (\underline{z} <= \underline{x} ⇒ ˙¬ ( ˙¬ (\underline{z} = \underline{x})n)n)n ⊢ Repetition ⊳ ˙¬ (\underline{z} <= \underline{x} ⇒ ˙¬ ( ˙¬ (\underline{z} = \underline{x})n)n)n ≫ ˙¬ (\underline{z} <= \underline{x} ⇒ ˙¬ ( ˙¬ (\underline{z} = \underline{x})n)n)n; FirstConjunct ⊳ ˙¬ (\underline{z} <= \underline{x} ⇒ ˙¬ ( ˙¬ (\underline{z} = \underline{x})n)n)n ≫ \underline{z} <= \underline{x}; subLeqRight ⊳ \underline{x} = \underline{y} ⊳ \underline{z} <= \underline{x} ≫ \underline{z} <= \underline{y}; SecondConjunct ⊳ ˙¬ (\underline{z} <= \underline{x} ⇒ ˙¬ ( ˙¬ (\underline{z} = \underline{x})n)n)n ≫ ˙¬ (\underline{z} = \underline{x})n; SubNeqRight ⊳ \underline{x} = \underline{y} ⊳ ˙¬ (\underline{z} = \underline{x})n ≫ ˙¬ (\underline{z} = \underline{y})n; JoinConjuncts ⊳ \underline{z} <= \underline{y} ⊳ ˙¬ (\underline{z} = \underline{y})n ≫ ˙¬ (\underline{z} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{z} = \underline{y})n)n)n⌉, p_{0}, c)]

[SubLessRight →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ ˙¬ (\underline{z} <= \underline{x} ⇒ ˙¬ ( ˙¬ (\underline{z} = \underline{x})n)n)n ⊢ ˙¬ (\underline{z} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{z} = \underline{y})n)n)n]

[SubLessRight →^{tex} “SubLessRight”]

[SubLessRight →^{pyk} “lemma subLessRight”]

SubLessLeft

[SubLessLeft →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ ˙¬ (\underline{x} <= \underline{z} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{z})n)n)n ⊢ Repetition ⊳ ˙¬ (\underline{x} <= \underline{z} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{z})n)n)n ≫ ˙¬ (\underline{x} <= \underline{z} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{z})n)n)n; FirstConjunct ⊳ ˙¬ (\underline{x} <= \underline{z} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{z})n)n)n ≫ \underline{x} <= \underline{z}; subLeqLeft ⊳ \underline{x} = \underline{y} ⊳ \underline{x} <= \underline{z} ≫ \underline{y} <= \underline{z}; SecondConjunct ⊳ ˙¬ (\underline{x} <= \underline{z} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{z})n)n)n ≫ ˙¬ (\underline{x} = \underline{z})n; SubNeqLeft ⊳ \underline{x} = \underline{y} ⊳ ˙¬ (\underline{x} = \underline{z})n ≫ ˙¬ (\underline{y} = \underline{z})n; JoinConjuncts ⊳ \underline{y} <= \underline{z} ⊳ ˙¬ (\underline{y} = \underline{z})n ≫ ˙¬ (\underline{y} <= \underline{z} ⇒ ˙¬ ( ˙¬ (\underline{y} = \underline{z})n)n)n⌉, p_{0}, c)]

[SubLessLeft →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ ˙¬ (\underline{x} <= \underline{z} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{z})n)n)n ⊢ ˙¬ (\underline{y} <= \underline{z} ⇒ ˙¬ ( ˙¬ (\underline{y} = \underline{z})n)n)n]

[SubLessLeft →^{tex} “SubLessLeft”]

[SubLessLeft →^{pyk} “lemma subLessLeft”]

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})) ⇒ ˙¬ ( ˙¬ (\underline{x} = (\underline{y} + (−u\underline{z})))n)n)n ⊢ NegativeToLeft(Less) ⊳ ˙¬ (\underline{x} <= (\underline{y} + (−u\underline{z})) ⇒ ˙¬ ( ˙¬ (\underline{x} = (\underline{y} + (−u\underline{z})))n)n)n ≫ ˙¬ ((\underline{x} + \underline{z}) <= \underline{y} ⇒ ˙¬ ( ˙¬ ((\underline{x} + \underline{z}) = \underline{y})n)n)n; plusCommutativity ≫ (\underline{x} + \underline{z}) = (\underline{z} + \underline{x}); SubLessLeft ⊳ (\underline{x} + \underline{z}) = (\underline{z} + \underline{x}) ⊳ ˙¬ ((\underline{x} + \underline{z}) <= \underline{y} ⇒ ˙¬ ( ˙¬ ((\underline{x} + \underline{z}) = \underline{y})n)n)n ≫ ˙¬ ((\underline{z} + \underline{x}) <= \underline{y} ⇒ ˙¬ ( ˙¬ ((\underline{z}+\underline{x}) = \underline{y})n)n)n; PositiveToRight(Less)⊳ ˙¬ ((\underline{z}+\underline{x}) <= \underline{y} ⇒ ˙¬ ( ˙¬ ((\underline{z}+\underline{x}) = \underline{y})n)n)n ≫ ˙¬ (\underline{z} <= (\underline{y} + (−u\underline{x})) ⇒ ˙¬ ( ˙¬ (\underline{z} = (\underline{y} + (−u\underline{x})))n)n)n⌉, p_{0}, c)]

[SwitchTerms(x < y − z) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ (\underline{x} <= (\underline{y} + (−u\underline{z})) ⇒ ˙¬ ( ˙¬ (\underline{x} = (\underline{y} + (−u\underline{z})))n)n)n ⊢ ˙¬ (\underline{z} <= (\underline{y} + (−u\underline{x})) ⇒ ˙¬ ( ˙¬ (\underline{z} = (\underline{y} + (−u\underline{x})))n)n)n]

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

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

SwitchTerms(x − y < z)

[SwitchTerms(x−y < z) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ ((\underline{x}+(−u\underline{y})) <= \underline{z} ⇒ ˙¬ ( ˙¬ ((\underline{x} + (−u\underline{y})) = \underline{z})n)n)n ⊢ NegativeToRight(Less) ⊳ ˙¬ ((\underline{x} + (−u\underline{y})) <= \underline{z} ⇒ ˙¬ ( ˙¬ ((\underline{x} + (−u\underline{y})) = \underline{z})n)n)n ≫ ˙¬ (\underline{x} <= (\underline{z} + \underline{y}) ⇒ ˙¬ ( ˙¬ (\underline{x} = (\underline{z} + \underline{y}))n)n)n; plusCommutativity ≫ (\underline{z} + \underline{y}) = (\underline{y} + \underline{z}); SubLessRight ⊳ (\underline{z} + \underline{y}) = (\underline{y} + \underline{z}) ⊳ ˙¬ (\underline{x} <= (\underline{z} + \underline{y}) ⇒ ˙¬ ( ˙¬ (\underline{x} = (\underline{z} + \underline{y}))n)n)n ≫ ˙¬ (\underline{x} <= (\underline{y} + \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = (\underline{y} + \underline{z}))n)n)n; PositiveToLeft(Less) ⊳ ˙¬ (\underline{x} <= (\underline{y} + \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = (\underline{y} + \underline{z}))n)n)n ≫ ˙¬ ((\underline{x} + (−u\underline{z})) <= \underline{y} ⇒ ˙¬ ( ˙¬ ((\underline{x} + (−u\underline{z})) = \underline{y})n)n)n⌉, p_{0}, c)]

[SwitchTerms(x − y < z) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ ((\underline{x} + (−u\underline{y})) <= \underline{z} ⇒ ˙¬ ( ˙¬ ((\underline{x} + (−u\underline{y})) = \underline{z})n)n)n ⊢ ˙¬ ((\underline{x} + (−u\underline{z})) <= \underline{y} ⇒ ˙¬ ( ˙¬ ((\underline{x} + (−u\underline{z})) = \underline{y})n)n)n]

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

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

LessAddition

[LessAddition →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⊢ LessLeq ⊳ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ≫ \underline{x} <= \underline{y}; leqAddition ⊳ \underline{x} <= \underline{y} ≫ (\underline{x} + \underline{z}) <= (\underline{y} + \underline{z}); LessNeq ⊳ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ≫ ˙¬ (\underline{x} = \underline{y})n; NeqAddition ⊳ ˙¬ (\underline{x} = \underline{y})n ≫ ˙¬ ((\underline{x} + \underline{z}) = (\underline{y} + \underline{z}))n; JoinConjuncts ⊳ (\underline{x} + \underline{z}) <= (\underline{y} + \underline{z}) ⊳ ˙¬ ((\underline{x} + \underline{z}) = (\underline{y} + \underline{z}))n ≫ ˙¬ ((\underline{x} + \underline{z}) <= (\underline{y} + \underline{z}) ⇒ ˙¬ ( ˙¬ ((\underline{x} + \underline{z}) = (\underline{y} + \underline{z}))n)n)n⌉, p_{0}, c)] [LessAddition →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⊢ ˙¬ ((\underline{x} + \underline{z}) <= (\underline{y} + \underline{z}) ⇒ ˙¬ ( ˙¬ ((\underline{x} + \underline{z}) = (\underline{y} + \underline{z}))n)n)n]

[LessAddition →^{tex} “LessAddition”]

[LessAddition →^{pyk} “lemma lessAddition”]

LessAdditionLeft

[LessAdditionLeft →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⊢ LessAddition ⊳ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ≫ ˙¬ ((\underline{x} + \underline{z}) <= (\underline{y} + \underline{z}) ⇒ ˙¬ ( ˙¬ ((\underline{x} + \underline{z}) = (\underline{y} + \underline{z}))n)n)n; plusCommutativity ≫ (\underline{x} + \underline{z}) = (\underline{z} + \underline{x}); SubLessLeft ⊳ (\underline{x} + \underline{z}) = (\underline{z} + \underline{x}) ⊳ ˙¬ ((\underline{x} + \underline{z}) <= (\underline{y} + \underline{z}) ⇒ ˙¬ ( ˙¬ ((\underline{x} + \underline{z}) = (\underline{y} + \underline{z}))n)n)n ≫ ˙¬ ((\underline{z} + \underline{x}) <= (\underline{y} + \underline{z}) ⇒ ˙¬ ( ˙¬ ((\underline{z} + \underline{x}) = (\underline{y} + \underline{z}))n)n)n; plusCommutativity ≫ (\underline{y} + \underline{z}) = (\underline{z} + \underline{y}); SubLessRight ⊳ (\underline{y} + \underline{z}) = (\underline{z} + \underline{y}) ⊳ ˙¬ ((\underline{z} + \underline{x}) <= (\underline{y} + \underline{z}) ⇒ ˙¬ ( ˙¬ ((\underline{z} + \underline{x}) = (\underline{y} + \underline{z}))n)n)n ≫ ˙¬ ((\underline{z} + \underline{x}) <= (\underline{z} + \underline{y}) ⇒ ˙¬ ( ˙¬ ((\underline{z} + \underline{x}) = (\underline{z} + \underline{y}))n)n)n⌉, p_{0}, c)]

[LessAdditionLeft →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⊢ ˙¬ ((\underline{z} + \underline{x}) <= (\underline{z} + \underline{y}) ⇒ ˙¬ ( ˙¬ ((\underline{z} + \underline{x}) = (\underline{z} + \underline{y}))n)n)n]

[LessAdditionLeft →^{tex} “LessAdditionLeft”]

[LessAdditionLeft →^{pyk} “lemma lessAdditionLeft”]

LessMultiplication

[LessMultiplication →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ (0 <= \underline{z} ⇒ ˙¬ ( ˙¬ (0 = \underline{z})n)n)n ⊢ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⊢ LessLeq ⊳ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ≫ \underline{x} <= \underline{y}; LessLeq ⊳ ˙¬ (0 <= \underline{z} ⇒ ˙¬ ( ˙¬ (0 = \underline{z})n)n)n ≫ 0 <= \underline{z}; leqMultiplication⊳0 <= \underline{z}⊳\underline{x} <= \underline{y} ≫ (\underline{x}∗\underline{z}) <= (\underline{y}∗\underline{z}); LessNeq⊳ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ≫ ˙¬ (\underline{x} = \underline{y})n; LessNeq ⊳ ˙¬ (0 <= \underline{z} ⇒ ˙¬ ( ˙¬ (0 = \underline{z})n)n)n ≫ ˙¬ (0 = \underline{z})n; NeqSymmetry⊳ ˙¬ (0 = \underline{z})n ≫ ˙¬ (\underline{z} = 0)n; NeqMultiplication⊳ ˙¬ (\underline{z} = 0)n⊳ ˙¬ (\underline{x} = \underline{y})n ≫ ˙¬ ((\underline{x}∗\underline{z}) = (\underline{y}∗\underline{z}))n; LeqNeqLess⊳(\underline{x}∗\underline{z}) <= (\underline{y}∗\underline{z})⊳ ˙¬ ((\underline{x}∗\underline{z}) = (\underline{y} ∗ \underline{z}))n ≫ ˙¬ ((\underline{x} ∗ \underline{z}) <= (\underline{y} ∗ \underline{z}) ⇒ ˙¬ ( ˙¬ ((\underline{x} ∗ \underline{z}) = (\underline{y} ∗ \underline{z}))n)n)n⌉, p_{0}, c)]

[LessMultiplication →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ (0 <= \underline{z} ⇒ ˙¬ ( ˙¬ (0 = \underline{z})n)n)n ⊢ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⊢ ˙¬ ((\underline{x} ∗ \underline{z}) <= (\underline{y} ∗ \underline{z}) ⇒ ˙¬ ( ˙¬ ((\underline{x} ∗ \underline{z}) = (\underline{y} ∗ \underline{z}))n)n)n]

[LessMultiplication →^{tex} “LessMultiplication”]

[LessMultiplication →^{pyk} “lemma lessMultiplication”]

LessMultiplicationLeft

[LessMultiplicationLeft →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ (0 <= \underline{z} ⇒ ˙¬ ( ˙¬ (0 = \underline{z})n)n)n ⊢ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⊢ LessMultiplication ⊳ ˙¬ (0 <= \underline{z} ⇒ ˙¬ ( ˙¬ (0 = \underline{z})n)n)n ⊳ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ≫ ˙¬ ((\underline{x} ∗ \underline{z}) <= (\underline{y} ∗ \underline{z}) ⇒ ˙¬ ( ˙¬ ((\underline{x} ∗ \underline{z}) = (\underline{y} ∗ \underline{z}))n)n)n; timesCommutativity ≫ (\underline{x} ∗ \underline{z}) = (\underline{z} ∗ \underline{x}); timesCommutativity ≫ (\underline{y} ∗ \underline{z}) = (\underline{z} ∗ \underline{y}); SubLessLeft ⊳ (\underline{x} ∗ \underline{z}) = (\underline{z} ∗ \underline{x}) ⊳ ˙¬ ((\underline{x} ∗ \underline{z}) <= (\underline{y} ∗ \underline{z}) ⇒ ˙¬ ( ˙¬ ((\underline{x} ∗ \underline{z}) = (\underline{y} ∗ \underline{z}))n)n)n ≫ ˙¬ ((\underline{z} ∗ \underline{x}) <= (\underline{y} ∗ \underline{z}) ⇒ ˙¬ ( ˙¬ ((\underline{z} ∗ \underline{x}) = (\underline{y}∗\underline{z}))n)n)n; SubLessRight⊳(\underline{y}∗\underline{z}) = (\underline{z}∗\underline{y})⊳ ˙¬ ((\underline{z}∗\underline{x}) <= (\underline{y}∗\underline{z}) ⇒ ˙¬ ( ˙¬ ((\underline{z}∗\underline{x}) = (\underline{y} ∗ \underline{z}))n)n)n ≫ ˙¬ ((\underline{z} ∗ \underline{x}) <= (\underline{z} ∗ \underline{y}) ⇒ ˙¬ ( ˙¬ ((\underline{z} ∗ \underline{x}) = (\underline{z} ∗ \underline{y}))n)n)n⌉, p_{0}, c)]

[LessMultiplicationLeft →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ (0 <= \underline{z} ⇒ ˙¬ ( ˙¬ (0 = \underline{z})n)n)n ⊢ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⊢ ˙¬ ((\underline{z} ∗ \underline{x}) <= (\underline{z} ∗ \underline{y}) ⇒ ˙¬ ( ˙¬ ((\underline{z} ∗ \underline{x}) = (\underline{z} ∗ \underline{y}))n)n)n]

[LessMultiplicationLeft →^{tex} “LessMultiplicationLeft”]

[LessMultiplicationLeft →^{pyk} “lemma lessMultiplicationLeft”]

LessDivision

[LessDivision →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: 0 <= \underline{z} ⊢ ˙¬ ((\underline{x} ∗ \underline{z}) <= (\underline{y} ∗ \underline{z}) ⇒ ˙¬ ( ˙¬ ((\underline{x} ∗ \underline{z}) = (\underline{y} ∗ \underline{z}))n)n)n ⊢ FromLess ⊳ ˙¬ ((\underline{x} ∗ \underline{z}) <= (\underline{y} ∗ \underline{z}) ⇒ ˙¬ ( ˙¬ ((\underline{x} ∗ \underline{z}) = (\underline{y} ∗ \underline{z}))n)n)n ≫ ˙¬ ((\underline{y} ∗ \underline{z}) <= (\underline{x} ∗ \underline{z}))n; leqMultiplicationAxiom ≫ 0 <= \underline{z} ⇒ \underline{y} <= \underline{x} ⇒ (\underline{y} ∗ \underline{z}) <= (\underline{x} ∗ \underline{z}); MP ⊳ 0 <= \underline{z} ⇒ \underline{y} <= \underline{x} ⇒ (\underline{y} ∗ \underline{z}) <= (\underline{x} ∗ \underline{z}) ⊳ 0 <= \underline{z} ≫ \underline{y} <= \underline{x} ⇒ (\underline{y} ∗ \underline{z}) <= (\underline{x} ∗ \underline{z}); Contrapositive ⊳ \underline{y} <= \underline{x} ⇒ (\underline{y} ∗ \underline{z}) <= (\underline{x} ∗ \underline{z}) ≫ ˙¬ ((\underline{y} ∗ \underline{z}) <= (\underline{x} ∗ \underline{z}))n ⇒ ˙¬ (\underline{y} <= \underline{x})n; MP ⊳ ˙¬ ((\underline{y} ∗ \underline{z}) <= (\underline{x} ∗ \underline{z}))n ⇒ ˙¬ (\underline{y} <= \underline{x})n ⊳ ˙¬ ((\underline{y} ∗ \underline{z}) <= (\underline{x} ∗ \underline{z}))n ≫ ˙¬ (\underline{y} <= \underline{x})n; ToLess ⊳ ˙¬ (\underline{y} <= \underline{x})n ≫ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n⌉, p_{0}, c)]

[LessDivision →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: 0 <= \underline{z} ⊢ ˙¬ ((\underline{x} ∗ \underline{z}) <= (\underline{y} ∗ \underline{z}) ⇒ ˙¬ ( ˙¬ ((\underline{x} ∗ \underline{z}) = (\underline{y} ∗ \underline{z}))n)n)n ⊢ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n]

[LessDivision →^{tex} “LessDivision”]

[LessDivision →^{pyk} “lemma lessDivision”]

PositiveToRight(Less)

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

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

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

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

PositiveToLeft(Less)

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

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

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

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

NegativeToLeft(Less)

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

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

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

NegativeToRight(Less)

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

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

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

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

AddEquations(Less)

[AddEquations(Less) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⊢ ˙¬ (\underline{z} <= \underline{u} ⇒ ˙¬ ( ˙¬ (\underline{z} = \underline{u})n)n)n ⊢ LessAddition ⊳ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ≫ ˙¬ ((\underline{x} + \underline{z}) <= (\underline{y} + \underline{z}) ⇒ ˙¬ ( ˙¬ ((\underline{x} + \underline{z}) = (\underline{y}+\underline{z}))n)n)n; LessAdditionLeft⊳ ˙¬ (\underline{z} <= \underline{u} ⇒ ˙¬ ( ˙¬ (\underline{z} = \underline{u})n)n)n ≫ ˙¬ ((\underline{y}+\underline{z}) <= (\underline{y} + \underline{u}) ⇒ ˙¬ ( ˙¬ ((\underline{y} + \underline{z}) = (\underline{y} + \underline{u}))n)n)n; LessTransitivity ⊳ ˙¬ ((\underline{x} + \underline{z}) <= (\underline{y} + \underline{z}) ⇒ ˙¬ ( ˙¬ ((\underline{x} + \underline{z}) = (\underline{y} + \underline{z}))n)n)n ⊳ ˙¬ ((\underline{y} + \underline{z}) <= (\underline{y} + \underline{u}) ⇒ ˙¬ ( ˙¬ ((\underline{y} + \underline{z}) = (\underline{y} + \underline{u}))n)n)n ≫ ˙¬ ((\underline{x} + \underline{z}) <= (\underline{y} + \underline{u}) ⇒ ˙¬ ( ˙¬ ((\underline{x} + \underline{z}) = (\underline{y} + \underline{u}))n)n)n⌉, p_{0}, c)]

[AddEquations(Less) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⊢ ˙¬ (\underline{z} <= \underline{u} ⇒ ˙¬ ( ˙¬ (\underline{z} = \underline{u})n)n)n ⊢ ˙¬ ((\underline{x} + \underline{z}) <= (\underline{y} + \underline{u}) ⇒ ˙¬ ( ˙¬ ((\underline{x} + \underline{z}) = (\underline{y} + \underline{u}))n)n)n]

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

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

AddEquations(LeqLess)

[AddEquations(LeqLess) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: \underline{x} <= \underline{y} ⊢ ˙¬ (\underline{z} <= \underline{u} ⇒ ˙¬ ( ˙¬ (\underline{z} = \underline{u})n)n)n ⊢ leqAddition ⊳ \underline{x} <= \underline{y} ≫ (\underline{x} + \underline{z}) <= (\underline{y} + \underline{z}); LessAdditionLeft ⊳ ˙¬ (\underline{z} <= \underline{u} ⇒ ˙¬ ( ˙¬ (\underline{z} = \underline{u})n)n)n ≫ ˙¬ ((\underline{y} + \underline{z}) <= (\underline{y} + \underline{u}) ⇒ ˙¬ ( ˙¬ ((\underline{y} + \underline{z}) = (\underline{y} + \underline{u}))n)n)n; leqLessTransitivity ⊳ (\underline{x} + \underline{z}) <= (\underline{y} + \underline{z}) ⊳ ˙¬ ((\underline{y} + \underline{z}) <= (\underline{y} + \underline{u}) ⇒ ˙¬ ( ˙¬ ((\underline{y} + \underline{z}) = (\underline{y} + \underline{u}))n)n)n ≫ ˙¬ ((\underline{x} + \underline{z}) <= (\underline{y} + \underline{u}) ⇒ ˙¬ ( ˙¬ ((\underline{x} + \underline{z}) = (\underline{y} + \underline{u}))n)n)n⌉, p_{0}, c)]

[AddEquations(LeqLess) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: \underline{x} <= \underline{y} ⊢ ˙¬ (\underline{z} <= \underline{u} ⇒ ˙¬ ( ˙¬ (\underline{z} = \underline{u})n)n)n ⊢ ˙¬ ((\underline{x} + \underline{z}) <= (\underline{y} + \underline{u}) ⇒ ˙¬ ( ˙¬ ((\underline{x} + \underline{z}) = (\underline{y} + \underline{u}))n)n)n]

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

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

reciprocalToLeft(Less)

[reciprocalToLeft(Less) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ (0 <= \underline{z} ⇒ ˙¬ ( ˙¬ (0 = \underline{z})n)n)n ⊢ ˙¬ (\underline{x} <= (\underline{y} ∗ rec\underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = (\underline{y} ∗ rec\underline{z}))n)n)n ⊢ LessMultiplication ⊳ ˙¬ (0 <= \underline{z} ⇒ ˙¬ ( ˙¬ (0 = \underline{z})n)n)n ⊳ ˙¬ (\underline{x} <= (\underline{y} ∗ rec\underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = (\underline{y} ∗ rec\underline{z}))n)n)n ≫ ˙¬ ((\underline{x} ∗ \underline{z}) <= ((\underline{y} ∗ rec\underline{z}) ∗ \underline{z}) ⇒ ˙¬ ( ˙¬ ((\underline{x} ∗ \underline{z}) = ((\underline{y} ∗ rec\underline{z}) ∗ \underline{z}))n)n)n; Three2threeFactors ≫ ((\underline{y} ∗ rec\underline{z}) ∗ \underline{z}) = ((\underline{y} ∗ \underline{z}) ∗ rec\underline{z}); PositiveNonzero ⊳ ˙¬ (0 <= \underline{z} ⇒ ˙¬ ( ˙¬ (0 = \underline{z})n)n)n ≫ ˙¬ (\underline{z} = 0)n; x = x ∗ y ∗ (1/y) ⊳ ˙¬ (\underline{z} = 0)n ≫ \underline{y} = ((\underline{y} ∗ \underline{z}) ∗ rec\underline{z}); eqSymmetry ⊳ \underline{y} = ((\underline{y} ∗ \underline{z}) ∗ rec\underline{z}) ≫ ((\underline{y} ∗ \underline{z}) ∗ rec\underline{z}) = \underline{y}; eqTransitivity ⊳ ((\underline{y} ∗ rec\underline{z}) ∗ \underline{z}) = ((\underline{y} ∗ \underline{z}) ∗ rec\underline{z}) ⊳ ((\underline{y} ∗ \underline{z}) ∗ rec\underline{z}) = \underline{y} ≫ ((\underline{y} ∗ rec\underline{z}) ∗ \underline{z}) = \underline{y}; SubLessRight⊳((\underline{y}∗rec\underline{z})∗\underline{z}) = \underline{y}⊳ ˙¬ ((\underline{x}∗\underline{z}) <= ((\underline{y}∗rec\underline{z})∗\underline{z}) ⇒ ˙¬ ( ˙¬ ((\underline{x}∗\underline{z}) = ((\underline{y} ∗ rec\underline{z}) ∗ \underline{z}))n)n)n ≫ ˙¬ ((\underline{x} ∗ \underline{z}) <= \underline{y} ⇒ ˙¬ ( ˙¬ ((\underline{x} ∗ \underline{z}) = \underline{y})n)n)n⌉, p_{0}, c)]

[reciprocalToLeft(Less) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ (0 <= \underline{z} ⇒ ˙¬ ( ˙¬ (0 = \underline{z})n)n)n ⊢ ˙¬ (\underline{x} <= (\underline{y} ∗ rec\underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = (\underline{y} ∗ rec\underline{z}))n)n)n ⊢ ˙¬ ((\underline{x} ∗ \underline{z}) <= \underline{y} ⇒ ˙¬ ( ˙¬ ((\underline{x} ∗ \underline{z}) = \underline{y})n)n)n]

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

[reciprocalToLeft(Less) →^{pyk} “lemma reciprocalToLeft(Less)”]

LessNegated

[LessNegated →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⊢ LessLeq ⊳ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ≫ \underline{x} <= \underline{y}; LeqNegated ⊳ \underline{x} <= \underline{y} ≫ (−u\underline{y}) <= (−u\underline{x}); LessNeq ⊳ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ≫ ˙¬ (\underline{x} = \underline{y})n; NeqNegated ⊳ ˙¬ (\underline{x} = \underline{y})n ≫ ˙¬ ((−u\underline{x}) = (−u\underline{y}))n; NeqSymmetry ⊳ ˙¬ ((−u\underline{x}) = (−u\underline{y}))n ≫ ˙¬ ((−u\underline{y}) = (−u\underline{x}))n; LeqNeqLess ⊳ (−u\underline{y}) <= (−u\underline{x}) ⊳ ˙¬ ((−u\underline{y}) = (−u\underline{x}))n ≫ ˙¬ ((−u\underline{y}) <= (−u\underline{x}) ⇒ ˙¬ ( ˙¬ ((−u\underline{y}) = (−u\underline{x}))n)n)n⌉, p_{0}, c)]

[LessNegated →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⊢ ˙¬ ((−u\underline{y}) <= (−u\underline{x}) ⇒ ˙¬ ( ˙¬ ((−u\underline{y}) = (−u\underline{x}))n)n)n]

[LessNegated →^{tex} “LessNegated”]

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

PositiveNonzero

[PositiveNonzero →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ⊢ Repetition ⊳ ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ≫ ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n; SecondConjunct ⊳ ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ≫ ˙¬ (0 = \underline{x})n; NeqSymmetry ⊳ ˙¬ (0 = \underline{x})n ≫ ˙¬ (\underline{x} = 0)n⌉, p_{0}, c)]

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

[PositiveNonzero →^{tex} “PositiveNonzero”]

[PositiveNonzero →^{pyk} “lemma positiveNonzero”]

PositiveNegated

[PositiveNegated →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ⊢ LessNegated ⊳ ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ≫ ˙¬ ((−u\underline{x}) <= (−u0) ⇒ ˙¬ ( ˙¬ ((−u\underline{x}) = (−u0))n)n)n; −0 = 0 ≫ (−u0) = 0; SubLessRight ⊳ (−u0) = 0 ⊳ ˙¬ ((−u\underline{x}) <= (−u0) ⇒ ˙¬ ( ˙¬ ((−u\underline{x}) = (−u0))n)n)n ≫ ˙¬ ((−u\underline{x}) <= 0 ⇒ ˙¬ ( ˙¬ ((−u\underline{x}) = 0)n)n)n⌉, p_{0}, c)]

[PositiveNegated →^{stmt} SystemQ ⊢ ∀\underline{x}: ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ⊢ ˙¬ ((−u\underline{x}) <= 0 ⇒ ˙¬ ( ˙¬ ((−u\underline{x}) = 0)n)n)n]

[PositiveNegated →^{tex} “PositiveNegated”]

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

NonpositiveNegated

[NonpositiveNegated →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: \underline{x} <= 0 ⊢ LeqNegated ⊳ \underline{x} <= 0 ≫ (−u0) <= (−u\underline{x}); −0 = 0 ≫ (−u0) = 0; subLeqLeft ⊳ (−u0) = 0 ⊳ (−u0) <= (−u\underline{x}) ≫ 0 <= (−u\underline{x})⌉, p_{0}, c)] [NonpositiveNegated →^{stmt} SystemQ ⊢ ∀\underline{x}: \underline{x} <= 0 ⊢ 0 <= (−u\underline{x})]

[NonpositiveNegated →^{tex} “NonpositiveNegated”]

[NonpositiveNegated →^{pyk} “lemma nonpositiveNegated”]

NegativeNegated

[NegativeNegated →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ˙¬ (\underline{x} <= 0 ⇒ ˙¬ ( ˙¬ (\underline{x} = 0)n)n)n ⊢ LessNegated ⊳ ˙¬ (\underline{x} <= 0 ⇒ ˙¬ ( ˙¬ (\underline{x} = 0)n)n)n ≫ ˙¬ ((−u0) <= (−u\underline{x}) ⇒ ˙¬ ( ˙¬ ((−u0) = (−u\underline{x}))n)n)n; −0 = 0 ≫ (−u0) = 0; SubLessLeft ⊳ (−u0) = 0 ⊳ ˙¬ ((−u0) <= (−u\underline{x}) ⇒ ˙¬ ( ˙¬ ((−u0) = (−u\underline{x}))n)n)n ≫ ˙¬ (0 <= (−u\underline{x}) ⇒ ˙¬ ( ˙¬ (0 = (−u\underline{x}))n)n)n⌉, p_{0}, c)]

[NegativeNegated →^{stmt} SystemQ ⊢ ∀\underline{x}: ˙¬ (\underline{x} <= 0 ⇒ ˙¬ ( ˙¬ (\underline{x} = 0)n)n)n ⊢ ˙¬ (0 <= (−u\underline{x}) ⇒ ˙¬ ( ˙¬ (0 = (−u\underline{x}))n)n)n]

[NegativeNegated →^{tex} “NegativeNegated”]

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

NonnegativeNegated

[NonnegativeNegated →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: 0 <= \underline{x} ⊢ LeqNegated ⊳ 0 <= \underline{x} ≫ (−u\underline{x}) <= (−u0); −0 = 0 ≫ (−u0) = 0; subLeqRight ⊳ (−u0) = 0 ⊳ (−u\underline{x}) <= (−u0) ≫ (−u\underline{x}) <= 0⌉, p_{0}, c)]

[NonnegativeNegated →^{stmt} SystemQ ⊢ ∀\underline{x}: 0 <= \underline{x} ⊢ (−u\underline{x}) <= 0]

[NonnegativeNegated →^{tex} “NonnegativeNegated”]

[NonnegativeNegated →^{pyk} “lemma nonnegativeNegated”]

PositiveHalved

[PositiveHalved →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ⊢ 0 < 1/2 ≫ ˙¬ (0 <= rec(1 + 1) ⇒ ˙¬ ( ˙¬ (0 = rec(1 + 1))n)n)n; LessMultiplicationLeft ⊳ ˙¬ (0 <= rec(1 + 1) ⇒ ˙¬ ( ˙¬ (0 = rec(1 + 1))n)n)n ⊳ ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ≫ ˙¬ ((rec(1 + 1) ∗ 0) <= (rec(1 + 1) ∗ \underline{x}) ⇒ ˙¬ ( ˙¬ ((rec(1 + 1) ∗ 0) = (rec(1 + 1) ∗ \underline{x}))n)n)n; x ∗ 0 = 0 ≫ (rec(1 + 1) ∗ 0) = 0; SubLessLeft ⊳ (rec(1 + 1) ∗ 0) = 0 ⊳ ˙¬ ((rec(1 + 1) ∗ 0) <= (rec(1 + 1) ∗ \underline{x}) ⇒ ˙¬ ( ˙¬ ((rec(1 + 1) ∗ 0) = (rec(1 + 1) ∗ \underline{x}))n)n)n ≫ ˙¬ (0 <= (rec(1 + 1) ∗ \underline{x}) ⇒ ˙¬ ( ˙¬ (0 = (rec(1 + 1) ∗ \underline{x}))n)n)n⌉, p_{0}, c)] [PositiveHalved →^{stmt} SystemQ ⊢ ∀\underline{x}: ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ⊢ ˙¬ (0 <= (rec(1 + 1) ∗ \underline{x}) ⇒ ˙¬ ( ˙¬ (0 = (rec(1 + 1) ∗ \underline{x}))n)n)n]

[PositiveHalved →^{tex} “PositiveHalved”]

[PositiveHalved →^{pyk} “lemma positiveHalved”]

PositiveInverted

[PositiveInverted →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ⊢ FirstConjunct ⊳ ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ≫ 0 <= \underline{x}; SecondConjunct ⊳ ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ≫ ˙¬ (0 = \underline{x})n; NeqSymmetry ⊳ ˙¬ (0 = \underline{x})n ≫ ˙¬ (\underline{x} = 0)n; 0 < 1 ≫ ˙¬ (0 <= 1 ⇒ ˙¬ ( ˙¬ (0 = 1)n)n)n; x ∗ 0 = 0 ≫ (\underline{x} ∗ 0) = 0; x ∗ y = zBackwards ⊳ (\underline{x} ∗ 0) = 0 ≫ 0 = (0 ∗ \underline{x}); SubLessLeft ⊳ 0 = (0 ∗ \underline{x}) ⊳ ˙¬ (0 <= 1 ⇒ ˙¬ ( ˙¬ (0 = 1)n)n)n ≫ ˙¬ ((0 ∗ \underline{x}) <= 1 ⇒ ˙¬ ( ˙¬ ((0 ∗ \underline{x}) = 1)n)n)n; Reciprocal ⊳ ˙¬ (\underline{x} = 0)n ≫ (\underline{x} ∗ rec\underline{x}) = 1; x ∗ y = zBackwards ⊳ (\underline{x} ∗ rec\underline{x}) = 1 ≫ 1 = (rec\underline{x} ∗ \underline{x}); SubLessRight ⊳ 1 = (rec\underline{x}∗\underline{x})⊳ ˙¬ ((0∗\underline{x}) <= 1 ⇒ ˙¬ ( ˙¬ ((0∗\underline{x}) = 1)n)n)n ≫ ˙¬ ((0∗\underline{x}) <= (rec\underline{x}∗\underline{x}) ⇒ ˙¬ ( ˙¬ ((0∗\underline{x}) = (rec\underline{x}∗\underline{x}))n)n)n; LessDivision⊳0 <= \underline{x}⊳ ˙¬ ((0∗\underline{x}) <= (rec\underline{x}∗\underline{x}) ⇒ ˙¬ ( ˙¬ ((0 ∗ \underline{x}) = (rec\underline{x} ∗ \underline{x}))n)n)n ≫ ˙¬ (0 <= rec\underline{x} ⇒ ˙¬ ( ˙¬ (0 = rec\underline{x})n)n)n⌉, p_{0}, c)]

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

[PositiveInverted →^{tex} “PositiveInverted”]

[PositiveInverted →^{pyk} “lemma positiveInverted”]

NonnegativeNumerical

[NonnegativeNumerical →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: 0 <= \underline{x} ⊢ Numerical ≫ ˙¬ ( ˙¬ (0 <= \underline{x} ⇒ ˙¬ (|\underline{x}| = \underline{x})n)n)n ⇒ ˙¬ ( ˙¬ (0 <= \underline{x})n ⇒ ˙¬ (|\underline{x}| = (−u\underline{x}))n)n; AddDoubleNeg ⊳ 0 <= \underline{x} ≫ ˙¬ ( ˙¬ (0 <= \underline{x})n)n; ToNegatedAnd(1) ⊳ ˙¬ ( ˙¬ (0 <= \underline{x})n)n ≫ ˙¬ ( ˙¬ ( ˙¬ (0 <= \underline{x})n ⇒ ˙¬ (|\underline{x}| = (−u\underline{x}))n)n)n; NegateDisjunct2 ⊳ ˙¬ ( ˙¬ (0 <= \underline{x} ⇒ ˙¬ (|\underline{x}| = \underline{x})n)n)n ⇒ ˙¬ ( ˙¬ (0 <= \underline{x})n ⇒ ˙¬ (|\underline{x}| = (−u\underline{x}))n)n ⊳ ˙¬ ( ˙¬ ( ˙¬ (0 <= \underline{x})n ⇒ ˙¬ (|\underline{x}| = (−u\underline{x}))n)n)n ≫ ˙¬ (0 <= \underline{x} ⇒ ˙¬ (|\underline{x}| = \underline{x})n)n; SecondConjunct ⊳ ˙¬ (0 <= \underline{x} ⇒ ˙¬ (|\underline{x}| = \underline{x})n)n ≫ |\underline{x}| = \underline{x}⌉, p_{0}, c)]

[NonnegativeNumerical →^{stmt} SystemQ ⊢ ∀\underline{x}: 0 <= \underline{x} ⊢ |\underline{x}| = \underline{x}]

[NonnegativeNumerical →^{tex} “NonnegativeNumerical”]

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

NegativeNumerical

[NegativeNumerical →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ˙¬ (\underline{x} <= 0 ⇒ ˙¬ ( ˙¬ (\underline{x} = 0)n)n)n ⊢ Numerical ≫ ˙¬ ( ˙¬ (0 <= \underline{x} ⇒ ˙¬ (|\underline{x}| = \underline{x})n)n)n ⇒ ˙¬ ( ˙¬ (0 <= \underline{x})n ⇒ ˙¬ (|\underline{x}| = (−u\underline{x}))n)n; FromLess ⊳ ˙¬ (\underline{x} <= 0 ⇒ ˙¬ ( ˙¬ (\underline{x} = 0)n)n)n ≫ ˙¬ (0 <= \underline{x})n; ToNegatedAnd(1) ⊳ ˙¬ (0 <= \underline{x})n ≫ ˙¬ ( ˙¬ (0 <= \underline{x} ⇒ ˙¬ (|\underline{x}| = \underline{x})n)n)n; NegateDisjunct1 ⊳ ˙¬ ( ˙¬ (0 <= \underline{x} ⇒ ˙¬ (|\underline{x}| = \underline{x})n)n)n ⇒ ˙¬ ( ˙¬ (0 <= \underline{x})n ⇒ ˙¬ (|\underline{x}| = (−u\underline{x}))n)n ⊳ ˙¬ ( ˙¬ (0 <= \underline{x} ⇒ ˙¬ (|\underline{x}| = \underline{x})n)n)n ≫ ˙¬ ( ˙¬ (0 <= \underline{x})n ⇒ ˙¬ (|\underline{x}| = (−u\underline{x}))n)n; SecondConjunct ⊳ ˙¬ ( ˙¬ (0 <= \underline{x})n ⇒ ˙¬ (|\underline{x}| = (−u\underline{x}))n)n ≫ |\underline{x}| = (−u\underline{x})⌉, p_{0}, c)]

[NegativeNumerical →^{stmt} SystemQ ⊢ ∀\underline{x}: ˙¬ (\underline{x} <= 0 ⇒ ˙¬ ( ˙¬ (\underline{x} = 0)n)n)n ⊢ |\underline{x}| = (−u\underline{x})]

[NegativeNumerical →^{tex} “NegativeNumerical”]

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

PositiveNumerical

[PositiveNumerical →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ⊢ LessLeq ⊳ ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ≫ 0 <= \underline{x}; NonnegativeNumerical ⊳ 0 <= \underline{x} ≫ |\underline{x}| = \underline{x}⌉, p_{0}, c)]

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

[PositiveNumerical →^{tex} “PositiveNumerical”]

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

lemma nonpositiveNumerical

[lemma nonpositiveNumerical →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ˙¬ (\underline{x} <= 0 ⇒ ˙¬ ( ˙¬ (\underline{x} = 0)n)n)n ⊢ NegativeNumerical ⊳ ˙¬ (\underline{x} <= 0 ⇒ ˙¬ ( ˙¬ (\underline{x} = 0)n)n)n ≫ |\underline{x}| = (−u\underline{x}); ∀\underline{x}: \underline{x} = 0 ⊢ eqSymmetry ⊳ \underline{x} = 0 ≫ 0 = \underline{x}; eqLeq ⊳ 0 = \underline{x} ≫ 0 <= \underline{x}; NonnegativeNumerical ⊳ 0 <= \underline{x} ≫ |\underline{x}| = \underline{x}; −0 = 0 ≫ (−u0) = 0; eqSymmetry ⊳ (−u0) = 0 ≫ 0 = (−u0); EqNegated ⊳ 0 = \underline{x} ≫ (−u0) = (−u\underline{x}); eqTransitivity5 ⊳ |\underline{x}| = \underline{x} ⊳ \underline{x} = 0 ⊳ 0 = (−u0) ⊳ (−u0) = (−u\underline{x}) ≫ |\underline{x}| = (−u\underline{x}); ∀\underline{x}: Ded ⊳ ∀\underline{x}: ˙¬ (\underline{x} <= 0 ⇒ ˙¬ ( ˙¬ (\underline{x} = 0)n)n)n ⊢ |\underline{x}| = (−u\underline{x}) ≫ ˙¬ (\underline{x} <= 0 ⇒ ˙¬ ( ˙¬ (\underline{x} = 0)n)n)n ⇒ |\underline{x}| = (−u\underline{x}); Ded ⊳ ∀\underline{x}: \underline{x} = 0 ⊢ |\underline{x}| = (−u\underline{x}) ≫ \underline{x} = 0 ⇒ |\underline{x}| = (−u\underline{x}); \underline{x} <= 0 ⊢ LeqLessEq ⊳ \underline{x} <= 0 ≫ ˙¬ ( ˙¬ (\underline{x} <= 0 ⇒ ˙¬ ( ˙¬ (\underline{x} = 0)n)n)n)n ⇒ \underline{x} = 0; FromDisjuncts ⊳ ˙¬ ( ˙¬ (\underline{x} <= 0 ⇒ ˙¬ ( ˙¬ (\underline{x} = 0)n)n)n)n ⇒ \underline{x} = 0 ⊳ ˙¬ (\underline{x} <= 0 ⇒ ˙¬ ( ˙¬ (\underline{x} = 0)n)n)n ⇒ |\underline{x}| = (−u\underline{x}) ⊳ \underline{x} = 0 ⇒ |\underline{x}| = (−u\underline{x}) ≫ |\underline{x}| = (−u\underline{x})⌉, p_{0}, c)] [lemma nonpositiveNumerical →^{stmt} SystemQ ⊢ ∀\underline{x}: \underline{x} <= 0 ⊢ |\underline{x}| = (−u\underline{x})]

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

|0| = 0

[|0| = 0 →^{proof} λc.λx.P(⌈SystemQ ⊢ leqReflexivity ≫ 0 <= 0; NonnegativeNumerical ⊳ 0 <= 0 ≫ |0| = 0⌉, p_{0}, c)]

[|0| = 0 →^{stmt} SystemQ ⊢ |0| = 0]

[|0| = 0 →^{tex} “|0|=0”]

[|0| = 0 →^{pyk} “lemma |0|=0”]

0 <= |x|

[0 <= |x| →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: 0 <= \underline{x} ⊢ NonnegativeNumerical⊳0 <= \underline{x} ≫ |\underline{x}| = \underline{x}; eqSymmetry ⊳ |\underline{x}| = \underline{x} ≫ \underline{x} = |\underline{x}|; subLeqRight ⊳ \underline{x} = |\underline{x}| ⊳ 0 <= \underline{x} ≫ 0 <= |\underline{x}|; ∀\underline{x}: ˙¬ (0 <= \underline{x})n ⊢ ToLess ⊳ ˙¬ (0 <= \underline{x})n ≫ ˙¬ (\underline{x} <= 0 ⇒ ˙¬ ( ˙¬ (\underline{x} = 0)n)n)n; NegativeNumerical ⊳ ˙¬ (\underline{x} <= 0 ⇒ ˙¬ ( ˙¬ (\underline{x} = 0)n)n)n ≫ |\underline{x}| = (−u\underline{x}); eqSymmetry ⊳ |\underline{x}| = (−u\underline{x}) ≫ (−u\underline{x}) = |\underline{x}|; NegativeNegated ⊳ ˙¬ (\underline{x} <= 0 ⇒ ˙¬ ( ˙¬ (\underline{x} = 0)n)n)n ≫ ˙¬ (0 <= (−u\underline{x}) ⇒ ˙¬ ( ˙¬ (0 = (−u\underline{x}))n)n)n; LessLeq ⊳ ˙¬ (0 <= (−u\underline{x}) ⇒ ˙¬ ( ˙¬ (0 = (−u\underline{x}))n)n)n ≫ 0 <= (−u\underline{x}); subLeqRight ⊳ (−u\underline{x}) = |\underline{x}| ⊳ 0 <= (−u\underline{x}) ≫ 0 <= |\underline{x}|; ∀\underline{x}: Ded ⊳ ∀\underline{x}: 0 <= \underline{x} ⊢ 0 <= |\underline{x}| ≫ 0 <= \underline{x} ⇒ 0 <= |\underline{x}|; Ded ⊳ ∀\underline{x}: ˙¬ (0 <= \underline{x})n ⊢ 0 <= |\underline{x}| ≫ ˙¬ (0 <= \underline{x})n ⇒ 0 <= |\underline{x}|; FromNegations ⊳ 0 <= \underline{x} ⇒ 0 <= |\underline{x}| ⊳ ˙¬ (0 <= \underline{x})n ⇒ 0 <= |\underline{x}| ≫ 0 <= |\underline{x}|⌉, p_{0}, c)]

[0 <= |x| →^{stmt} SystemQ ⊢ ∀\underline{x}: 0 <= |\underline{x}|]

[0 <= |x| →^{tex} “0<=|x|”]

[0 <= |x| →^{pyk} “lemma 0<=|x|”]

x <= |x|

[x <= |x| →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: 0 <= \underline{x} ⊢ NonnegativeNumerical ≫ |\underline{x}| = \underline{x}; eqSymmetry ⊳ |\underline{x}| = \underline{x} ≫ \underline{x} = |\underline{x}|; eqLeq ⊳ \underline{x} = |\underline{x}| ≫ \underline{x} <= |\underline{x}|; ∀\underline{x}: \underline{x} <= 0 ⊢ 0 <= |x| ≫ 0 <= |\underline{x}|; leqTransitivity ⊳ \underline{x} <= 0 ⊳ 0 <= |\underline{x}| ≫ \underline{x} <= |\underline{x}|; ∀\underline{x}: Ded ⊳ ∀\underline{x}: 0 <= \underline{x} ⊢ \underline{x} <= |\underline{x}| ≫ 0 <= \underline{x} ⇒ \underline{x} <= |\underline{x}|; Ded ⊳ ∀\underline{x}: \underline{x} <= 0 ⊢ \underline{x} <= |\underline{x}| ≫ \underline{x} <= 0 ⇒ \underline{x} <= |\underline{x}|; FromLeqGeq ⊳ 0 <= \underline{x} ⇒ \underline{x} <= |\underline{x}| ⊳ \underline{x} <= 0 ⇒ \underline{x} <= |\underline{x}| ≫ \underline{x} <= |\underline{x}|⌉, p_{0}, c)] [x <= |x| →^{stmt} SystemQ ⊢ ∀\underline{x}: \underline{x} <= |\underline{x}|]

[x <= |x| →^{tex} “x<=|x|”]

[x <= |x| →^{pyk} “lemma x<=|x|”]

FromPositiveNumerical

[FromPositiveNumerical →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: 0 <= \underline{x} ⊢ ˙¬ (0 <= |\underline{x}| ⇒ ˙¬ ( ˙¬ (0 = |\underline{x}|)n)n)n ⊢ NonnegativeNumerical ⊳ 0 <= \underline{x} ≫ |\underline{x}| = \underline{x}; SubLessRight ⊳ |\underline{x}| = \underline{x} ⊳ ˙¬ (0 <= |\underline{x}| ⇒ ˙¬ ( ˙¬ (0 = |\underline{x}|)n)n)n ≫ ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n; LessNeq ⊳ ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ≫ ˙¬ (0 = \underline{x})n; NeqSymmetry ⊳ ˙¬ (0 = \underline{x})n ≫ ˙¬ (\underline{x} = 0)n; ∀\underline{x}: \underline{x} <= 0 ⊢ ˙¬ (0 <= |\underline{x}| ⇒ ˙¬ ( ˙¬ (0 = |\underline{x}|)n)n)n ⊢ lemma nonpositiveNumerical ⊳ \underline{x} <= 0 ≫ |\underline{x}| = (−u\underline{x}); SubLessRight ⊳ |\underline{x}| = (−u\underline{x}) ⊳ ˙¬ (0 <= |\underline{x}| ⇒ ˙¬ ( ˙¬ (0 = |\underline{x}|)n)n)n ≫ ˙¬ (0 <= (−u\underline{x}) ⇒ ˙¬ ( ˙¬ (0 = (−u\underline{x}))n)n)n; PositiveNegated ⊳ ˙¬ (0 <= (−u\underline{x}) ⇒ ˙¬ ( ˙¬ (0 = (−u\underline{x}))n)n)n ≫ ˙¬ ((−u(−u\underline{x})) <= 0 ⇒ ˙¬ ( ˙¬ ((−u(−u\underline{x})) = 0)n)n)n; DoubleMinus ≫ (−u(−u\underline{x})) = \underline{x}; SubLessLeft ⊳ (−u(−u\underline{x})) = \underline{x} ⊳ ˙¬ ((−u(−u\underline{x})) <= 0 ⇒ ˙¬ ( ˙¬ ((−u(−u\underline{x})) = 0)n)n)n ≫ ˙¬ (\underline{x} <= 0 ⇒ ˙¬ ( ˙¬ (\underline{x} = 0)n)n)n; LessNeq ⊳ ˙¬ (\underline{x} <= 0 ⇒ ˙¬ ( ˙¬ (\underline{x} = 0)n)n)n ≫ ˙¬ (\underline{x} = 0)n; ∀\underline{x}: Ded ⊳ ∀\underline{x}: 0 <= \underline{x} ⊢ ˙¬ (0 <= |\underline{x}| ⇒ ˙¬ ( ˙¬ (0 = |\underline{x}|)n)n)n ⊢ ˙¬ (\underline{x} = 0)n ≫ 0 <= \underline{x} ⇒ ˙¬ (0 <= |\underline{x}| ⇒ ˙¬ ( ˙¬ (0 = |\underline{x}|)n)n)n ⇒ ˙¬ (\underline{x} = 0)n; Ded ⊳ ∀\underline{x}: \underline{x} <= 0 ⊢ ˙¬ (0 <= |\underline{x}| ⇒ ˙¬ ( ˙¬ (0 = |\underline{x}|)n)n)n ⊢ ˙¬ (\underline{x} = 0)n ≫ \underline{x} <= 0 ⇒ ˙¬ (0 <= |\underline{x}| ⇒ ˙¬ ( ˙¬ (0 = |\underline{x}|)n)n)n ⇒ ˙¬ (\underline{x} = 0)n; ˙¬ (0 <= |\underline{x}| ⇒ ˙¬ ( ˙¬ (0 = |\underline{x}|)n)n)n ⊢ FromLeqGeq ⊳ 0 <= \underline{x} ⇒ ˙¬ (0 <= |\underline{x}| ⇒ ˙¬ ( ˙¬ (0 = |\underline{x}|)n)n)n ⇒ ˙¬ (\underline{x} = 0)n ⊳ \underline{x} <= 0 ⇒ ˙¬ (0 <= |\underline{x}| ⇒ ˙¬ ( ˙¬ (0 = |\underline{x}|)n)n)n ⇒ ˙¬ (\underline{x} = 0)n ≫ ˙¬ (0 <= |\underline{x}| ⇒ ˙¬ ( ˙¬ (0 = |\underline{x}|)n)n)n ⇒ ˙¬ (\underline{x} = 0)n; MP ⊳ ˙¬ (0 <= |\underline{x}| ⇒ ˙¬ ( ˙¬ (0 = |\underline{x}|)n)n)n ⇒ ˙¬ (\underline{x} = 0)n ⊳ ˙¬ (0 <= |\underline{x}| ⇒ ˙¬ ( ˙¬ (0 = |\underline{x}|)n)n)n ≫ ˙¬ (\underline{x} = 0)n⌉, p_{0}, c)]

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

[FromPositiveNumerical →^{tex} “FromPositiveNumerical”]

[FromPositiveNumerical →^{pyk} “lemma fromPositiveNumerical”]

SameNumerical

[SameNumerical →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ \underline{x} = \underline{y} ⊢ NonnegativeNumerical ⊳ 0 <= \underline{x} ≫ |\underline{x}| = \underline{x}; subLeqRight ⊳ \underline{x} = \underline{y} ⊳ 0 <= \underline{x} ≫ 0 <= \underline{y}; NonnegativeNumerical ⊳ 0 <= \underline{y} ≫ |\underline{y}| = \underline{y}; eqSymmetry ⊳ |\underline{y}| = \underline{y} ≫ \underline{y} = |\underline{y}|; eqTransitivity4 ⊳ |\underline{x}| = \underline{x} ⊳ \underline{x} = \underline{y} ⊳ \underline{y} = |\underline{y}| ≫ |\underline{x}| = |\underline{y}|; ∀\underline{x}: ∀\underline{y}: ˙¬ (0 <= \underline{x})n ⊢ \underline{x} = \underline{y} ⊢ ToLess ⊳ ˙¬ (0 <= \underline{x})n ≫ ˙¬ (\underline{x} <= 0 ⇒ ˙¬ ( ˙¬ (\underline{x} = 0)n)n)n; NegativeNumerical ⊳ ˙¬ (\underline{x} <= 0 ⇒ ˙¬ ( ˙¬ (\underline{x} = 0)n)n)n ≫ |\underline{x}| = (−u\underline{x}); SubLessLeft ⊳ \underline{x} = \underline{y} ⊳ ˙¬ (\underline{x} <= 0 ⇒ ˙¬ ( ˙¬ (\underline{x} = 0)n)n)n ≫ ˙¬ (\underline{y} <= 0 ⇒ ˙¬ ( ˙¬ (\underline{y} = 0)n)n)n; NegativeNumerical ⊳ ˙¬ (\underline{y} <= 0 ⇒ ˙¬ ( ˙¬ (\underline{y} = 0)n)n)n ≫ |\underline{y}| = (−u\underline{y}); eqSymmetry⊳|\underline{y}| = (−u\underline{y}) ≫ (−u\underline{y}) = |\underline{y}|; EqNegated⊳\underline{x} = \underline{y} ≫ (−u\underline{x}) = (−u\underline{y}); eqTransitivity4 ⊳ |\underline{x}| = (−u\underline{x}) ⊳ (−u\underline{x}) = (−u\underline{y}) ⊳ (−u\underline{y}) = |\underline{y}| ≫ |\underline{x}| = |\underline{y}|; ∀\underline{x}: ∀\underline{y}: \underline{x} = \underline{y} ⊢ Ded ⊳ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ \underline{x} = \underline{y} ⊢ |\underline{x}| = |\underline{y}| ≫ 0 <= \underline{x} ⇒ \underline{x} = \underline{y} ⇒ |\underline{x}| = |\underline{y}|; Ded⊳∀\underline{x}: ∀\underline{y}: ˙¬ (0 <= \underline{x})n ⊢ \underline{x} = \underline{y} ⊢ |\underline{x}| = |\underline{y}| ≫ ˙¬ (0 <= \underline{x})n ⇒ \underline{x} = \underline{y} ⇒ |\underline{x}| = |\underline{y}|; FromNegations⊳0 <= \underline{x} ⇒ \underline{x} = \underline{y} ⇒ |\underline{x}| = |\underline{y}|⊳ ˙¬ (0 <= \underline{x})n ⇒ \underline{x} = \underline{y} ⇒ |\underline{x}| = |\underline{y}| ≫ \underline{x} = \underline{y} ⇒ |\underline{x}| = |\underline{y}|; MP ⊳ \underline{x} = \underline{y} ⇒ |\underline{x}| = |\underline{y}| ⊳ \underline{x} = \underline{y} ≫ |\underline{x}| = |\underline{y}|⌉, p_{0}, c)]

[SameNumerical →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} = \underline{y} ⊢ |\underline{x}| = |\underline{y}|]

[SameNumerical →^{tex} “SameNumerical”]

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

SignNumerical(+)

[SignNumerical(+) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ⊢ PositiveNumerical ⊳ ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ≫ |\underline{x}| = \underline{x}; PositiveNegated ⊳ ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ≫ ˙¬ ((−u\underline{x}) <= 0 ⇒ ˙¬ ( ˙¬ ((−u\underline{x}) = 0)n)n)n; NegativeNumerical ⊳ ˙¬ ((−u\underline{x}) <= 0 ⇒ ˙¬ ( ˙¬ ((−u\underline{x}) = 0)n)n)n ≫ |(−u\underline{x})| = (−u(−u\underline{x})); DoubleMinus ≫ (−u(−u\underline{x})) = \underline{x}; eqTransitivity ⊳ |(−u\underline{x})| = (−u(−u\underline{x})) ⊳ (−u(−u\underline{x})) = \underline{x} ≫ |(−u\underline{x})| = \underline{x}; eqSymmetry ⊳ |(−u\underline{x})| = \underline{x} ≫ \underline{x} = |(−u\underline{x})|; eqTransitivity ⊳ |\underline{x}| = \underline{x} ⊳ \underline{x} = |(−u\underline{x})| ≫ |\underline{x}| = |(−u\underline{x})|⌉, p_{0}, c)]

[SignNumerical(+) →^{stmt} SystemQ ⊢ ∀\underline{x}: ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ⊢ |\underline{x}| = |(−u\underline{x})|]

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

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

SignNumerical

[SignNumerical →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ˙¬ (\underline{x} <= 0 ⇒ ˙¬ ( ˙¬ (\underline{x} = 0)n)n)n ⊢ NegativeNegated ⊳ ˙¬ (\underline{x} <= 0 ⇒ ˙¬ ( ˙¬ (\underline{x} = 0)n)n)n ≫ ˙¬ (0 <= (−u\underline{x}) ⇒ ˙¬ ( ˙¬ (0 = (−u\underline{x}))n)n)n; SignNumerical(+) ⊳ ˙¬ (0 <= (−u\underline{x}) ⇒ ˙¬ ( ˙¬ (0 = (−u\underline{x}))n)n)n ≫ |(−u\underline{x})| = |(−u(−u\underline{x}))|; DoubleMinus ≫ (−u(−u\underline{x})) = \underline{x}; SameNumerical ⊳ (−u(−u\underline{x})) = \underline{x} ≫ |(−u(−u\underline{x}))| = |\underline{x}|; eqTransitivity ⊳ |(−u\underline{x})| = |(−u(−u\underline{x}))| ⊳ |(−u(−u\underline{x}))| = |\underline{x}| ≫ |(−u\underline{x})| = |\underline{x}|; eqSymmetry ⊳ |(−u\underline{x})| = |\underline{x}| ≫ |\underline{x}| = |(−u\underline{x})|; ∀\underline{x}: \underline{x} = 0 ⊢ EqNegated ⊳ \underline{x} = 0 ≫ (−u\underline{x}) = (−u0); −0 = 0 ≫ (−u0) = 0; eqSymmetry ⊳ \underline{x} = 0 ≫ 0 = \underline{x}; eqTransitivity4 ⊳ (−u\underline{x}) = (−u0) ⊳ (−u0) = 0 ⊳ 0 = \underline{x} ≫ (−u\underline{x}) = \underline{x}; eqSymmetry ⊳ (−u\underline{x}) = \underline{x} ≫ \underline{x} = (−u\underline{x}); SameNumerical ⊳ \underline{x} = (−u\underline{x}) ≫ |\underline{x}| = |(−u\underline{x})|; ∀\underline{x}: ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ⊢ SignNumerical(+) ⊳ ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ≫ |\underline{x}| = |(−u\underline{x})|; ∀\underline{x}: Ded ⊳ ∀\underline{x}: ˙¬ (\underline{x} <= 0 ⇒ ˙¬ ( ˙¬ (\underline{x} = 0)n)n)n ⊢ |\underline{x}| = |(−u\underline{x})| ≫ ˙¬ (\underline{x} <= 0 ⇒ ˙¬ ( ˙¬ (\underline{x} = 0)n)n)n ⇒ |\underline{x}| = |(−u\underline{x})|; Ded ⊳ ∀\underline{x}: \underline{x} = 0 ⊢ |\underline{x}| = |(−u\underline{x})| ≫ \underline{x} = 0 ⇒ |\underline{x}| = |(−u\underline{x})|; Ded ⊳ ∀\underline{x}: ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ⊢ |\underline{x}| = |(−u\underline{x})| ≫ ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ⇒ |\underline{x}| = |(−u\underline{x})|; LessTotality ≫ ˙¬ ( ˙¬ (\underline{x} <= 0 ⇒ ˙¬ ( ˙¬ (\underline{x} = 0)n)n)n)n ⇒ ˙¬ (\underline{x} = 0)n ⇒ ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n; From3Disjuncts ⊳ ˙¬ ( ˙¬ (\underline{x} <= 0 ⇒ ˙¬ ( ˙¬ (\underline{x} = 0)n)n)n)n ⇒ ˙¬ (\underline{x} = 0)n ⇒ ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ⊳ ˙¬ (\underline{x} <= 0 ⇒ ˙¬ ( ˙¬ (\underline{x} = 0)n)n)n ⇒ |\underline{x}| = |(−u\underline{x})| ⊳ \underline{x} = 0 ⇒ |\underline{x}| = |(−u\underline{x})| ⊳ ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ⇒ |\underline{x}| = |(−u\underline{x})| ≫ |\underline{x}| = |(−u\underline{x})|⌉, p_{0}, c)]

[SignNumerical →^{stmt} SystemQ ⊢ ∀\underline{x}: |\underline{x}| = |(−u\underline{x})|]

[SignNumerical →^{tex} “SignNumerical”]

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

ToNumericalLess

[ToNumericalLess →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⊢ 0 <= \underline{x} ⊢ NonnegativeNumerical ⊳ 0 <= \underline{x} ≫ |\underline{x}| = \underline{x}; eqSymmetry ⊳ |\underline{x}| = \underline{x} ≫ \underline{x} = |\underline{x}|; SubLessLeft ⊳ \underline{x} = |\underline{x}| ⊳ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ≫ ˙¬ (|\underline{x}| <= \underline{y} ⇒ ˙¬ ( ˙¬ (|\underline{x}| = \underline{y})n)n)n; ∀\underline{x}: ∀\underline{y}: ˙¬ ((−u\underline{y}) <= \underline{x} ⇒ ˙¬ ( ˙¬ ((−u\underline{y}) = \underline{x})n)n)n ⊢ \underline{x} <= 0 ⊢ LessNegated ⊳ ˙¬ ((−u\underline{y}) <= \underline{x} ⇒ ˙¬ ( ˙¬ ((−u\underline{y}) = \underline{x})n)n)n ≫ ˙¬ ((−u\underline{x}) <= (−u(−u\underline{y})) ⇒ ˙¬ ( ˙¬ ((−u\underline{x}) = (−u(−u\underline{y})))n)n)n; lemma nonpositiveNumerical ⊳ \underline{x} <= 0 ≫ |\underline{x}| = (−u\underline{x}); eqSymmetry ⊳ |\underline{x}| = (−u\underline{x}) ≫ (−u\underline{x}) = |\underline{x}|; SubLessLeft ⊳ (−u\underline{x}) = |\underline{x}| ⊳ ˙¬ ((−u\underline{x}) <= (−u(−u\underline{y})) ⇒ ˙¬ ( ˙¬ ((−u\underline{x}) = (−u(−u\underline{y})))n)n)n ≫ ˙¬ (|\underline{x}| <= (−u(−u\underline{y})) ⇒ ˙¬ ( ˙¬ (|\underline{x}| = (−u(−u\underline{y})))n)n)n; DoubleMinus ≫ (−u(−u\underline{y})) = \underline{y}; SubLessRight ⊳ (−u(−u\underline{y})) = \underline{y} ⊳ ˙¬ (|\underline{x}| <= (−u(−u\underline{y})) ⇒ ˙¬ ( ˙¬ (|\underline{x}| = (−u(−u\underline{y})))n)n)n ≫ ˙¬ (|\underline{x}| <= \underline{y} ⇒ ˙¬ ( ˙¬ (|\underline{x}| = \underline{y})n)n)n; ∀\underline{x}: ∀\underline{y}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⊢ 0 <= \underline{x} ⊢ ˙¬ (|\underline{x}| <= \underline{y} ⇒ ˙¬ ( ˙¬ (|\underline{x}| = \underline{y})n)n)n ≫ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⇒ 0 <= \underline{x} ⇒ ˙¬ (|\underline{x}| <= \underline{y} ⇒ ˙¬ ( ˙¬ (|\underline{x}| = \underline{y})n)n)n; Ded ⊳ ∀\underline{x}: ∀\underline{y}: ˙¬ ((−u\underline{y}) <= \underline{x} ⇒ ˙¬ ( ˙¬ ((−u\underline{y}) = \underline{x})n)n)n ⊢ \underline{x} <= 0 ⊢ ˙¬ (|\underline{x}| <= \underline{y} ⇒ ˙¬ ( ˙¬ (|\underline{x}| = \underline{y})n)n)n ≫ ˙¬ ((−u\underline{y}) <= \underline{x} ⇒ ˙¬ ( ˙¬ ((−u\underline{y}) = \underline{x})n)n)n ⇒ \underline{x} <= 0 ⇒ ˙¬ (|\underline{x}| <= \underline{y} ⇒ ˙¬ ( ˙¬ (|\underline{x}| = \underline{y})n)n)n; ˙¬ ((−u\underline{y}) <= \underline{x} ⇒ ˙¬ ( ˙¬ ((−u\underline{y}) = \underline{x})n)n)n ⊢ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⊢ MP ⊳ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⇒ 0 <= \underline{x} ⇒ ˙¬ (|\underline{x}| <= \underline{y} ⇒ ˙¬ ( ˙¬ (|\underline{x}| = \underline{y})n)n)n ⊳ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ≫ 0 <= \underline{x} ⇒ ˙¬ (|\underline{x}| <= \underline{y} ⇒ ˙¬ ( ˙¬ (|\underline{x}| = \underline{y})n)n)n; MP ⊳ ˙¬ ((−u\underline{y}) <= \underline{x} ⇒ ˙¬ ( ˙¬ ((−u\underline{y}) = \underline{x})n)n)n ⇒ \underline{x} <= 0 ⇒ ˙¬ (|\underline{x}| <= \underline{y} ⇒ ˙¬ ( ˙¬ (|\underline{x}| = \underline{y})n)n)n ⊳ ˙¬ ((−u\underline{y}) <= \underline{x} ⇒ ˙¬ ( ˙¬ ((−u\underline{y}) = \underline{x})n)n)n ≫ \underline{x} <= 0 ⇒ ˙¬ (|\underline{x}| <= \underline{y} ⇒ ˙¬ ( ˙¬ (|\underline{x}| = \underline{y})n)n)n; FromLeqGeq⊳0 <= \underline{x} ⇒ ˙¬ (|\underline{x}| <= \underline{y} ⇒ ˙¬ ( ˙¬ (|\underline{x}| = \underline{y})n)n)n⊳\underline{x} <= 0 ⇒ ˙¬ (|\underline{x}| <= \underline{y} ⇒ ˙¬ ( ˙¬ (|\underline{x}| = \underline{y})n)n)n ≫ ˙¬ (|\underline{x}| <= \underline{y} ⇒ ˙¬ ( ˙¬ (|\underline{x}| = \underline{y})n)n)n⌉, p_{0}, c)]

[ToNumericalLess →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ ((−u\underline{y}) <= \underline{x} ⇒ ˙¬ ( ˙¬ ((−u\underline{y}) = \underline{x})n)n)n ⊢ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⊢ ˙¬ (|\underline{x}| <= \underline{y} ⇒ ˙¬ ( ˙¬ (|\underline{x}| = \underline{y})n)n)n]

[ToNumericalLess →^{tex} “ToNumericalLess”]

[ToNumericalLess →^{pyk} “lemma toNumericalLess”]

FromNumericalGreater

[FromNumericalGreater →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ (\underline{x} <= |\underline{y}| ⇒ ˙¬ ( ˙¬ (\underline{x} = |\underline{y}|)n)n)n ⊢ 0 <= \underline{y} ⊢ NonnegativeNumerical ⊳ 0 <= \underline{y} ≫ |\underline{y}| = \underline{y}; SubLessRight ⊳ |\underline{y}| = \underline{y} ⊳ ˙¬ (\underline{x} <= |\underline{y}| ⇒ ˙¬ ( ˙¬ (\underline{x} = |\underline{y}|)n)n)n ≫ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n; WeakenOr1 ⊳ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ≫ ˙¬ ( ˙¬ (\underline{y} <= (−u\underline{x}) ⇒ ˙¬ ( ˙¬ (\underline{y} = (−u\underline{x}))n)n)n)n ⇒ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n; ∀\underline{x}: ∀\underline{y}: ˙¬ (\underline{x} <= |\underline{y}| ⇒ ˙¬ ( ˙¬ (\underline{x} = |\underline{y}|)n)n)n ⊢ \underline{y} <= 0 ⊢ lemma nonpositiveNumerical ⊳ \underline{y} <= 0 ≫ |\underline{y}| = (−u\underline{y}); SubLessRight ⊳ |\underline{y}| = (−u\underline{y}) ⊳ ˙¬ (\underline{x} <= |\underline{y}| ⇒ ˙¬ ( ˙¬ (\underline{x} = |\underline{y}|)n)n)n ≫ ˙¬ (\underline{x} <= (−u\underline{y}) ⇒ ˙¬ ( ˙¬ (\underline{x} = (−u\underline{y}))n)n)n; LessNegated ⊳ ˙¬ (\underline{x} <= (−u\underline{y}) ⇒ ˙¬ ( ˙¬ (\underline{x} = (−u\underline{y}))n)n)n ≫ ˙¬ ((−u(−u\underline{y})) <= (−u\underline{x}) ⇒ ˙¬ ( ˙¬ ((−u(−u\underline{y})) = (−u\underline{x}))n)n)n; DoubleMinus ≫ (−u(−u\underline{y})) = \underline{y}; SubLessLeft ⊳ (−u(−u\underline{y})) = \underline{y} ⊳ ˙¬ ((−u(−u\underline{y})) <= (−u\underline{x}) ⇒ ˙¬ ( ˙¬ ((−u(−u\underline{y})) = (−u\underline{x}))n)n)n ≫ ˙¬ (\underline{y} <= (−u\underline{x}) ⇒ ˙¬ ( ˙¬ (\underline{y} = (−u\underline{x}))n)n)n; WeakenOr2 ⊳ ˙¬ (\underline{y} <= (−u\underline{x}) ⇒ ˙¬ ( ˙¬ (\underline{y} = (−u\underline{x}))n)n)n ≫ ˙¬ ( ˙¬ (\underline{y} <= (−u\underline{x}) ⇒ ˙¬ ( ˙¬ (\underline{y} = (−u\underline{x}))n)n)n)n ⇒ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n; ∀\underline{x}: ∀\underline{y}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: ˙¬ (\underline{x} <= |\underline{y}| ⇒ ˙¬ ( ˙¬ (\underline{x} = |\underline{y}|)n)n)n ⊢ 0 <= \underline{y} ⊢ ˙¬ ( ˙¬ (\underline{y} <= (−u\underline{x}) ⇒ ˙¬ ( ˙¬ (\underline{y} = (−u\underline{x}))n)n)n)n ⇒ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ≫ ˙¬ (\underline{x} <= |\underline{y}| ⇒ ˙¬ ( ˙¬ (\underline{x} = |\underline{y}|)n)n)n ⇒ 0 <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{y} <= (−u\underline{x}) ⇒ ˙¬ ( ˙¬ (\underline{y} = (−u\underline{x}))n)n)n)n ⇒ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n; Ded ⊳ ∀\underline{x}: ∀\underline{y}: ˙¬ (\underline{x} <= |\underline{y}| ⇒ ˙¬ ( ˙¬ (\underline{x} = |\underline{y}|)n)n)n ⊢ \underline{y} <= 0 ⊢ ˙¬ ( ˙¬ (\underline{y} <= (−u\underline{x}) ⇒ ˙¬ ( ˙¬ (\underline{y} = (−u\underline{x}))n)n)n)n ⇒ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ≫ ˙¬ (\underline{x} <= |\underline{y}| ⇒ ˙¬ ( ˙¬ (\underline{x} = |\underline{y}|)n)n)n ⇒ \underline{y} <= 0 ⇒ ˙¬ ( ˙¬ (\underline{y} <= (−u\underline{x}) ⇒ ˙¬ ( ˙¬ (\underline{y} = (−u\underline{x}))n)n)n)n ⇒ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n; ˙¬ (\underline{x} <= |\underline{y}| ⇒ ˙¬ ( ˙¬ (\underline{x} = |\underline{y}|)n)n)n ⊢ MP ⊳ ˙¬ (\underline{x} <= |\underline{y}| ⇒ ˙¬ ( ˙¬ (\underline{x} = |\underline{y}|)n)n)n ⇒ 0 <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{y} <= (−u\underline{x}) ⇒ ˙¬ ( ˙¬ (\underline{y} = (−u\underline{x}))n)n)n)n ⇒ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⊳ ˙¬ (\underline{x} <= |\underline{y}| ⇒ ˙¬ ( ˙¬ (\underline{x} = |\underline{y}|)n)n)n ≫ 0 <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{y} <= (−u\underline{x}) ⇒ ˙¬ ( ˙¬ (\underline{y} = (−u\underline{x}))n)n)n)n ⇒ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n; MP ⊳ ˙¬ (\underline{x} <= |\underline{y}| ⇒ ˙¬ ( ˙¬ (\underline{x} = |\underline{y}|)n)n)n ⇒ \underline{y} <= 0 ⇒ ˙¬ ( ˙¬ (\underline{y} <= (−u\underline{x}) ⇒ ˙¬ ( ˙¬ (\underline{y} = (−u\underline{x}))n)n)n)n ⇒ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⊳ ˙¬ (\underline{x} <= |\underline{y}| ⇒ ˙¬ ( ˙¬ (\underline{x} = |\underline{y}|)n)n)n ≫ \underline{y} <= 0 ⇒ ˙¬ ( ˙¬ (\underline{y} <= (−u\underline{x}) ⇒ ˙¬ ( ˙¬ (\underline{y} = (−u\underline{x}))n)n)n)n ⇒ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n; FromLeqGeq ⊳ 0 <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{y} <= (−u\underline{x}) ⇒ ˙¬ ( ˙¬ (\underline{y} = (−u\underline{x}))n)n)n)n ⇒ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⊳ \underline{y} <= 0 ⇒ ˙¬ ( ˙¬ (\underline{y} <= (−u\underline{x}) ⇒ ˙¬ ( ˙¬ (\underline{y} = (−u\underline{x}))n)n)n)n ⇒ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ≫ ˙¬ ( ˙¬ (\underline{y} <= (−u\underline{x}) ⇒ ˙¬ ( ˙¬ (\underline{y} = (−u\underline{x}))n)n)n)n ⇒ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n⌉, p_{0}, c)]

[FromNumericalGreater →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ (\underline{x} <= |\underline{y}| ⇒ ˙¬ ( ˙¬ (\underline{x} = |\underline{y}|)n)n)n ⊢ ˙¬ ( ˙¬ (\underline{y} <= (−u\underline{x}) ⇒ ˙¬ ( ˙¬ (\underline{y} = (−u\underline{x}))n)n)n)n ⇒ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n]

[FromNumericalGreater →^{tex} “FromNumericalGreater”]

[FromNumericalGreater →^{pyk} “lemma fromNumericalGreater”]

NumericalDifference

[NumericalDifference →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: SignNumerical ≫ |(\underline{x} + (−u\underline{y}))| = |(−u(\underline{x} + (−u\underline{y})))|; MinusNegated ≫ (−u(\underline{x} + (−u\underline{y}))) = (\underline{y} + (−u\underline{x})); SameNumerical ⊳ (−u(\underline{x} + (−u\underline{y}))) = (\underline{y} + (−u\underline{x})) ≫ |(−u(\underline{x} + (−u\underline{y})))| = |(\underline{y} + (−u\underline{x}))|; eqTransitivity ⊳ |(\underline{x} + (−u\underline{y}))| = |(−u(\underline{x} + (−u\underline{y})))| ⊳ |(−u(\underline{x} + (−u\underline{y})))| = |(\underline{y} + (−u\underline{x}))| ≫ |(\underline{x} + (−u\underline{y}))| = |(\underline{y} + (−u\underline{x}))|⌉, p_{0}, c)]

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

[NumericalDifference →^{tex} “NumericalDifference”]

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

NumericalDifferenceLess(Helper)

[NumericalDifferenceLess(Helper) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: 0 <= (\underline{x} + (−u\underline{y})) ⊢ ˙¬ ((\underline{x} + (−u\underline{y})) <= \underline{z} ⇒ ˙¬ ( ˙¬ ((\underline{x} + (−u\underline{y})) = \underline{z})n)n)n ⊢ leqLessTransitivity ⊳ 0 <= (\underline{x} + (−u\underline{y})) ⊳ ˙¬ ((\underline{x} + (−u\underline{y})) <= \underline{z} ⇒ ˙¬ ( ˙¬ ((\underline{x} + (−u\underline{y})) = \underline{z})n)n)n ≫ ˙¬ (0 <= \underline{z} ⇒ ˙¬ ( ˙¬ (0 = \underline{z})n)n)n; PositiveNegated ⊳ ˙¬ (0 <= \underline{z} ⇒ ˙¬ ( ˙¬ (0 = \underline{z})n)n)n ≫ ˙¬ ((−u\underline{z}) <= 0 ⇒ ˙¬ ( ˙¬ ((−u\underline{z}) = 0)n)n)n; LessAdditionLeft ⊳ ˙¬ ((−u\underline{z}) <= 0 ⇒ ˙¬ ( ˙¬ ((−u\underline{z}) = 0)n)n)n ≫ ˙¬ ((\underline{y} + (−u\underline{z})) <= (\underline{y} + 0) ⇒ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) = (\underline{y}+0))n)n)n; plus0 ≫ (\underline{y}+0) = \underline{y}; SubLessRight⊳(\underline{y}+0) = \underline{y}⊳ ˙¬ ((\underline{y}+(−u\underline{z})) <= (\underline{y} + 0) ⇒ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) = (\underline{y} + 0))n)n)n ≫ ˙¬ ((\underline{y} + (−u\underline{z})) <= \underline{y} ⇒ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) = \underline{y})n)n)n; negativeToLeft(Leq)(1term) ⊳ 0 <= (\underline{x} + (−u\underline{y})) ≫ \underline{y} <= \underline{x}; LessLeqTransitivity ⊳ ˙¬ ((\underline{y} + (−u\underline{z})) <= \underline{y} ⇒ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) = \underline{y})n)n)n ⊳ \underline{y} <= \underline{x} ≫ ˙¬ ((\underline{y} + (−u\underline{z})) <= \underline{x} ⇒ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) = \underline{x})n)n)n; NegativeToRight(Less) ⊳ ˙¬ ((\underline{x} + (−u\underline{y})) <= \underline{z} ⇒ ˙¬ ( ˙¬ ((\underline{x} + (−u\underline{y})) = \underline{z})n)n)n ≫ ˙¬ (\underline{x} <= (\underline{z} + \underline{y}) ⇒ ˙¬ ( ˙¬ (\underline{x} = (\underline{z} + \underline{y}))n)n)n; plusCommutativity ≫ (\underline{z} + \underline{y}) = (\underline{y} + \underline{z}); SubLessRight ⊳ (\underline{z} + \underline{y}) = (\underline{y} + \underline{z}) ⊳ ˙¬ (\underline{x} <= (\underline{z} + \underline{y}) ⇒ ˙¬ ( ˙¬ (\underline{x} = (\underline{z} + \underline{y}))n)n)n ≫ ˙¬ (\underline{x} <= (\underline{y} + \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = (\underline{y} + \underline{z}))n)n)n; JoinConjuncts ⊳ ˙¬ ((\underline{y} + (−u\underline{z})) <= \underline{x} ⇒ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) = \underline{x})n)n)n ⊳ ˙¬ (\underline{x} <= (\underline{y} + \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = (\underline{y} + \underline{z}))n)n)n ≫ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) <= \underline{x} ⇒ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) = \underline{x})n)n)n ⇒ ˙¬ ( ˙¬ (\underline{x} <= (\underline{y} + \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = (\underline{y} + \underline{z}))n)n)n)n)n⌉, p_{0}, c)]

[NumericalDifferenceLess(Helper) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: 0 <= (\underline{x}+(−u\underline{y})) ⊢ ˙¬ ((\underline{x} + (−u\underline{y})) <= \underline{z} ⇒ ˙¬ ( ˙¬ ((\underline{x} + (−u\underline{y})) = \underline{z})n)n)n ⊢ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) <= \underline{x} ⇒ ˙¬ ( ˙¬ ((\underline{y}+(−u\underline{z})) = \underline{x})n)n)n ⇒ ˙¬ ( ˙¬ (\underline{x} <= (\underline{y}+\underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = (\underline{y}+\underline{z}))n)n)n)n)n]

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

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

NumericalDifferenceLess

[NumericalDifferenceLess →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ (|(\underline{x} + (−u\underline{y}))| <= \underline{z} ⇒ ˙¬ ( ˙¬ (|(\underline{x} + (−u\underline{y}))| = \underline{z})n)n)n ⊢ 0 <= (\underline{x} + (−u\underline{y})) ⊢ NonnegativeNumerical ⊳ 0 <= (\underline{x} + (−u\underline{y})) ≫ |(\underline{x} + (−u\underline{y}))| = (\underline{x} + (−u\underline{y})); SubLessLeft ⊳ |(\underline{x} + (−u\underline{y}))| = (\underline{x} + (−u\underline{y})) ⊳ ˙¬ (|(\underline{x} + (−u\underline{y}))| <= \underline{z} ⇒ ˙¬ ( ˙¬ (|(\underline{x} + (−u\underline{y}))| = \underline{z})n)n)n ≫ ˙¬ ((\underline{x} + (−u\underline{y})) <= \underline{z} ⇒ ˙¬ ( ˙¬ ((\underline{x} + (−u\underline{y})) = \underline{z})n)n)n; NumericalDifferenceLess(Helper) ⊳ 0 <= (\underline{x} + (−u\underline{y})) ⊳ ˙¬ ((\underline{x} + (−u\underline{y})) <= \underline{z} ⇒ ˙¬ ( ˙¬ ((\underline{x} + (−u\underline{y})) = \underline{z})n)n)n ≫ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) <= \underline{x} ⇒ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) = \underline{x})n)n)n ⇒ ˙¬ ( ˙¬ (\underline{x} <= (\underline{y} + \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = (\underline{y} + \underline{z}))n)n)n)n)n; ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ (|(\underline{x} + (−u\underline{y}))| <= \underline{z} ⇒ ˙¬ ( ˙¬ (|(\underline{x} + (−u\underline{y}))| = \underline{z})n)n)n ⊢ ˙¬ (0 <= (\underline{x} + (−u\underline{y})))n ⊢ ToLess ⊳ ˙¬ (0 <= (\underline{x} + (−u\underline{y})))n ≫ ˙¬ ((\underline{x} + (−u\underline{y})) <= 0 ⇒ ˙¬ ( ˙¬ ((\underline{x} + (−u\underline{y})) = 0)n)n)n; NegativeNumerical ⊳ ˙¬ ((\underline{x} + (−u\underline{y})) <= 0 ⇒ ˙¬ ( ˙¬ ((\underline{x} + (−u\underline{y})) = 0)n)n)n ≫ |(\underline{x} + (−u\underline{y}))| = (−u(\underline{x} + (−u\underline{y}))); MinusNegated ≫ (−u(\underline{x} + (−u\underline{y}))) = (\underline{y} + (−u\underline{x})); eqTransitivity ⊳ |(\underline{x} + (−u\underline{y}))| = (−u(\underline{x} + (−u\underline{y}))) ⊳ (−u(\underline{x} + (−u\underline{y}))) = (\underline{y} + (−u\underline{x})) ≫ |(\underline{x} + (−u\underline{y}))| = (\underline{y} + (−u\underline{x})); SubLessLeft ⊳ |(\underline{x} + (−u\underline{y}))| = (\underline{y} + (−u\underline{x})) ⊳ ˙¬ (|(\underline{x} + (−u\underline{y}))| <= \underline{z} ⇒ ˙¬ ( ˙¬ (|(\underline{x} + (−u\underline{y}))| = \underline{z})n)n)n ≫ ˙¬ ((\underline{y} + (−u\underline{x})) <= \underline{z} ⇒ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{x})) = \underline{z})n)n)n; NegativeNegated ⊳ ˙¬ ((\underline{x} + (−u\underline{y})) <= 0 ⇒ ˙¬ ( ˙¬ ((\underline{x} + (−u\underline{y})) = 0)n)n)n ≫ ˙¬ (0 <= (−u(\underline{x} + (−u\underline{y}))) ⇒ ˙¬ ( ˙¬ (0 = (−u(\underline{x}+(−u\underline{y}))))n)n)n; SubLessRight⊳(−u(\underline{x}+(−u\underline{y}))) = (\underline{y}+(−u\underline{x}))⊳ ˙¬ (0 <= (−u(\underline{x} + (−u\underline{y}))) ⇒ ˙¬ ( ˙¬ (0 = (−u(\underline{x} + (−u\underline{y}))))n)n)n ≫ ˙¬ (0 <= (\underline{y} + (−u\underline{x})) ⇒ ˙¬ ( ˙¬ (0 = (\underline{y} + (−u\underline{x})))n)n)n; LessLeq ⊳ ˙¬ (0 <= (\underline{y} + (−u\underline{x})) ⇒ ˙¬ ( ˙¬ (0 = (\underline{y}+(−u\underline{x})))n)n)n ≫ 0 <= (\underline{y}+(−u\underline{x})); NumericalDifferenceLess(Helper)⊳0 <= (\underline{y} + (−u\underline{x})) ⊳ ˙¬ ((\underline{y} + (−u\underline{x})) <= \underline{z} ⇒ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{x})) = \underline{z})n)n)n ≫ ˙¬ ( ˙¬ ((\underline{x} + (−u\underline{z})) <= \underline{y} ⇒ ˙¬ ( ˙¬ ((\underline{x} + (−u\underline{z})) = \underline{y})n)n)n ⇒ ˙¬ ( ˙¬ (\underline{y} <= (\underline{x} + \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{y} = (\underline{x} + \underline{z}))n)n)n)n)n; FirstConjunct ⊳ ˙¬ ( ˙¬ ((\underline{x} + (−u\underline{z})) <= \underline{y} ⇒ ˙¬ ( ˙¬ ((\underline{x} + (−u\underline{z})) = \underline{y})n)n)n ⇒ ˙¬ ( ˙¬ (\underline{y} <= (\underline{x} + \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{y} = (\underline{x} + \underline{z}))n)n)n)n)n ≫ ˙¬ ((\underline{x} + (−u\underline{z})) <= \underline{y} ⇒ ˙¬ ( ˙¬ ((\underline{x} + (−u\underline{z})) = \underline{y})n)n)n; NegativeToRight(Less) ⊳ ˙¬ ((\underline{x} + (−u\underline{z})) <= \underline{y} ⇒ ˙¬ ( ˙¬ ((\underline{x} + (−u\underline{z})) = \underline{y})n)n)n ≫ ˙¬ (\underline{x} <= (\underline{y} + \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = (\underline{y} + \underline{z}))n)n)n; SecondConjunct ⊳ ˙¬ ( ˙¬ ((\underline{x} + (−u\underline{z})) <= \underline{y} ⇒ ˙¬ ( ˙¬ ((\underline{x} + (−u\underline{z})) = \underline{y})n)n)n ⇒ ˙¬ ( ˙¬ (\underline{y} <= (\underline{x} + \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{y} = (\underline{x} + \underline{z}))n)n)n)n)n ≫ ˙¬ (\underline{y} <= (\underline{x} + \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{y} = (\underline{x} + \underline{z}))n)n)n; PositiveToLeft(Less) ⊳ ˙¬ (\underline{y} <= (\underline{x} + \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{y} = (\underline{x} + \underline{z}))n)n)n ≫ ˙¬ ((\underline{y} + (−u\underline{z})) <= \underline{x} ⇒ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) = \underline{x})n)n)n; JoinConjuncts ⊳ ˙¬ ((\underline{y} + (−u\underline{z})) <= \underline{x} ⇒ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) = \underline{x})n)n)n ⊳ ˙¬ (\underline{x} <= (\underline{y} + \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = (\underline{y} + \underline{z}))n)n)n ≫ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) <= \underline{x} ⇒ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) = \underline{x})n)n)n ⇒ ˙¬ ( ˙¬ (\underline{x} <= (\underline{y} + \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = (\underline{y} + \underline{z}))n)n)n)n)n; ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ (|(\underline{x} + (−u\underline{y}))| <= \underline{z} ⇒ ˙¬ ( ˙¬ (|(\underline{x} + (−u\underline{y}))| = \underline{z})n)n)n ⊢ 0 <= (\underline{x} + (−u\underline{y})) ⊢ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) <= \underline{x} ⇒ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) = \underline{x})n)n)n ⇒ ˙¬ ( ˙¬ (\underline{x} <= (\underline{y} + \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = (\underline{y} + \underline{z}))n)n)n)n)n ≫ ˙¬ (|(\underline{x} + (−u\underline{y}))| <= \underline{z} ⇒ ˙¬ ( ˙¬ (|(\underline{x} + (−u\underline{y}))| = \underline{z})n)n)n ⇒ 0 <= (\underline{x} + (−u\underline{y})) ⇒ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) <= \underline{x} ⇒ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) = \underline{x})n)n)n ⇒ ˙¬ ( ˙¬ (\underline{x} <= (\underline{y} + \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = (\underline{y} + \underline{z}))n)n)n)n)n; Ded ⊳ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ (|(\underline{x} + (−u\underline{y}))| <= \underline{z} ⇒ ˙¬ ( ˙¬ (|(\underline{x} + (−u\underline{y}))| = \underline{z})n)n)n ⊢ ˙¬ (0 <= (\underline{x} + (−u\underline{y})))n ⊢ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) <= \underline{x} ⇒ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) = \underline{x})n)n)n ⇒ ˙¬ ( ˙¬ (\underline{x} <= (\underline{y} + \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = (\underline{y} + \underline{z}))n)n)n)n)n ≫ ˙¬ (|(\underline{x} + (−u\underline{y}))| <= \underline{z} ⇒ ˙¬ ( ˙¬ (|(\underline{x} + (−u\underline{y}))| = \underline{z})n)n)n ⇒ ˙¬ (0 <= (\underline{x} + (−u\underline{y})))n ⇒ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) <= \underline{x} ⇒ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) = \underline{x})n)n)n ⇒ ˙¬ ( ˙¬ (\underline{x} <= (\underline{y}+\underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = (\underline{y}+\underline{z}))n)n)n)n)n; ˙¬ (|(\underline{x}+(−u\underline{y}))| <= \underline{z} ⇒ ˙¬ ( ˙¬ (|(\underline{x} + (−u\underline{y}))| = \underline{z})n)n)n ⊢ MP ⊳ ˙¬ (|(\underline{x} + (−u\underline{y}))| <= \underline{z} ⇒ ˙¬ ( ˙¬ (|(\underline{x} + (−u\underline{y}))| = \underline{z})n)n)n ⇒ 0 <= (\underline{x} + (−u\underline{y})) ⇒ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) <= \underline{x} ⇒ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) = \underline{x})n)n)n ⇒ ˙¬ ( ˙¬ (\underline{x} <= (\underline{y} + \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = (\underline{y} + \underline{z}))n)n)n)n)n ⊳ ˙¬ (|(\underline{x} + (−u\underline{y}))| <= \underline{z} ⇒ ˙¬ ( ˙¬ (|(\underline{x} + (−u\underline{y}))| = \underline{z})n)n)n ≫ 0 <= (\underline{x} + (−u\underline{y})) ⇒ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) <= \underline{x} ⇒ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) = \underline{x})n)n)n ⇒ ˙¬ ( ˙¬ (\underline{x} <= (\underline{y} +\underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = (\underline{y} +\underline{z}))n)n)n)n)n; MP⊳ ˙¬ (|(\underline{x}+(−u\underline{y}))| <= \underline{z} ⇒ ˙¬ ( ˙¬ (|(\underline{x} + (−u\underline{y}))| = \underline{z})n)n)n ⇒ ˙¬ (0 <= (\underline{x} + (−u\underline{y})))n ⇒ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) <= \underline{x} ⇒ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) = \underline{x})n)n)n ⇒ ˙¬ ( ˙¬ (\underline{x} <= (\underline{y} + \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = (\underline{y} + \underline{z}))n)n)n)n)n ⊳ ˙¬ (|(\underline{x} + (−u\underline{y}))| <= \underline{z} ⇒ ˙¬ ( ˙¬ (|(\underline{x} + (−u\underline{y}))| = \underline{z})n)n)n ≫ ˙¬ (0 <= (\underline{x} + (−u\underline{y})))n ⇒ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) <= \underline{x} ⇒ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) = \underline{x})n)n)n ⇒ ˙¬ ( ˙¬ (\underline{x} <= (\underline{y} + \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = (\underline{y} + \underline{z}))n)n)n)n)n; FromNegations ⊳ 0 <= (\underline{x} + (−u\underline{y})) ⇒ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) <= \underline{x} ⇒ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) = \underline{x})n)n)n ⇒ ˙¬ ( ˙¬ (\underline{x} <= (\underline{y} + \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = (\underline{y} + \underline{z}))n)n)n)n)n ⊳ ˙¬ (0 <= (\underline{x} + (−u\underline{y})))n ⇒ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) <= \underline{x} ⇒ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) = \underline{x})n)n)n ⇒ ˙¬ ( ˙¬ (\underline{x} <= (\underline{y} + \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = (\underline{y} + \underline{z}))n)n)n)n)n ≫ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) <= \underline{x} ⇒ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) = \underline{x})n)n)n ⇒ ˙¬ ( ˙¬ (\underline{x} <= (\underline{y} + \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = (\underline{y} + \underline{z}))n)n)n)n)n⌉, p_{0}, c)]

[NumericalDifferenceLess →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ (|(\underline{x} + (−u\underline{y}))| <= \underline{z} ⇒ ˙¬ ( ˙¬ (|(\underline{x} + (−u\underline{y}))| = \underline{z})n)n)n ⊢ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) <= \underline{x} ⇒ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) = \underline{x})n)n)n ⇒ ˙¬ ( ˙¬ (\underline{x} <= (\underline{y} + \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = (\underline{y} + \underline{z}))n)n)n)n)n]

[NumericalDifferenceLess →^{tex} “NumericalDifferenceLess”]

[NumericalDifferenceLess →^{pyk} “lemma numericalDifferenceLess”]

SplitNumericalSumHelper

[SplitNumericalSumHelper →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: |((−u\underline{x}) + (−u\underline{y}))| <= (|(−u\underline{x})| + |(−u\underline{y})|) ⊢ SignNumerical ≫ |\underline{x}| = |(−u\underline{x})|; SignNumerical ≫ |\underline{y}| = |(−u\underline{y})|; AddEquations ⊳ |\underline{x}| = |(−u\underline{x})| ⊳ |\underline{y}| = |(−u\underline{y})| ≫ (|\underline{x}| + |\underline{y}|) = (|(−u\underline{x})| + |(−u\underline{y})|); eqSymmetry ⊳ (|\underline{x}| + |\underline{y}|) = (|(−u\underline{x})| + |(−u\underline{y})|) ≫ (|(−u\underline{x})| + |(−u\underline{y})|) = (|\underline{x}| + |\underline{y}|); −x − y = −(x + y) ≫ ((−u\underline{x}) + (−u\underline{y})) = (−u(\underline{x} + \underline{y})); SameNumerical ⊳ ((−u\underline{x}) + (−u\underline{y})) = (−u(\underline{x} + \underline{y})) ≫ |((−u\underline{x}) + (−u\underline{y}))| = |(−u(\underline{x} + \underline{y}))|; SignNumerical ≫ |(\underline{x} + \underline{y})| = |(−u(\underline{x} + \underline{y}))|; eqSymmetry ⊳ |(\underline{x} + \underline{y})| = |(−u(\underline{x} + \underline{y}))| ≫ |(−u(\underline{x} + \underline{y}))| = |(\underline{x} + \underline{y})|; eqTransitivity ⊳ |((−u\underline{x}) + (−u\underline{y}))| = |(−u(\underline{x} + \underline{y}))| ⊳ |(−u(\underline{x} + \underline{y}))| = |(\underline{x} + \underline{y})| ≫ |((−u\underline{x}) + (−u\underline{y}))| = |(\underline{x} + \underline{y})|; subLeqRight ⊳ (|(−u\underline{x})| + |(−u\underline{y})|) = (|\underline{x}| + |\underline{y}|) ⊳ |((−u\underline{x}) + (−u\underline{y}))| <= (|(−u\underline{x})| + |(−u\underline{y})|) ≫ |((−u\underline{x}) + (−u\underline{y}))| <= (|\underline{x}| + |\underline{y}|); subLeqLeft ⊳ |((−u\underline{x}) + (−u\underline{y}))| = |(\underline{x} + \underline{y})| ⊳ |((−u\underline{x}) + (−u\underline{y}))| <= (|\underline{x}| + |\underline{y}|) ≫ |(\underline{x} + \underline{y})| <= (|\underline{x}| + |\underline{y}|)⌉, p_{0}, c)]

[SplitNumericalSumHelper →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: |((−u\underline{x}) + (−u\underline{y}))| <= (|(−u\underline{x})| + |(−u\underline{y})|) ⊢ |(\underline{x} + \underline{y})| <= (|\underline{x}| + |\underline{y}|)]

[SplitNumericalSumHelper →^{tex} “SplitNumericalSumHelper”]

[SplitNumericalSumHelper →^{pyk} “lemma splitNumericalSumHelper”]

splitNumericalSum(++)

[splitNumericalSum(++) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ 0 <= \underline{y} ⊢ AddEquations(Leq) ⊳ 0 <= \underline{x} ⊳ 0 <= \underline{y} ≫ (0 + 0) <= (\underline{x} + \underline{y}); plus0 ≫ (0 + 0) = 0; subLeqLeft ⊳ (0 + 0) = 0 ⊳ (0 + 0) <= (\underline{x} + \underline{y}) ≫ 0 <= (\underline{x} + \underline{y}); NonnegativeNumerical ⊳ 0 <= (\underline{x} + \underline{y}) ≫ |(\underline{x} + \underline{y})| = (\underline{x} + \underline{y}); NonnegativeNumerical ⊳ 0 <= \underline{x} ≫ |\underline{x}| = \underline{x}; NonnegativeNumerical ⊳ 0 <= \underline{y} ≫ |\underline{y}| = \underline{y}; AddEquations ⊳ |\underline{x}| = \underline{x} ⊳ |\underline{y}| = \underline{y} ≫ (|\underline{x}| + |\underline{y}|) = (\underline{x} + \underline{y}); eqSymmetry ⊳ (|\underline{x}| + |\underline{y}|) = (\underline{x} + \underline{y}) ≫ (\underline{x} + \underline{y}) = (|\underline{x}| + |\underline{y}|); eqTransitivity ⊳ |(\underline{x} + \underline{y})| = (\underline{x} + \underline{y}) ⊳ (\underline{x} + \underline{y}) = (|\underline{x}| + |\underline{y}|) ≫ |(\underline{x} + \underline{y})| = (|\underline{x}| + |\underline{y}|); eqLeq ⊳ |(\underline{x} + \underline{y})| = (|\underline{x}| + |\underline{y}|) ≫ |(\underline{x} + \underline{y})| <= (|\underline{x}| + |\underline{y}|)⌉, p_{0}, c)]

[splitNumericalSum(++) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ 0 <= \underline{y} ⊢ |(\underline{x} + \underline{y})| <= (|\underline{x}| + |\underline{y}|)]

[splitNumericalSum(++) →^{tex} “splitNumericalSum(++)”]

[splitNumericalSum(++) →^{pyk} “lemma splitNumericalSum(++)”]

splitNumericalSum(−−)

[splitNumericalSum(−−) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= 0 ⊢ \underline{y} <= 0 ⊢ NonpositiveNegated ⊳ \underline{x} <= 0 ≫ 0 <= (−u\underline{x}); NonpositiveNegated ⊳ \underline{y} <= 0 ≫ 0 <= (−u\underline{y}); splitNumericalSum(++) ⊳ 0 <= (−u\underline{x}) ⊳ 0 <= (−u\underline{y}) ≫ |((−u\underline{x}) + (−u\underline{y}))| <= (|(−u\underline{x})| + |(−u\underline{y})|); SplitNumericalSumHelper ⊳ |((−u\underline{x}) + (−u\underline{y}))| <= (|(−u\underline{x})| + |(−u\underline{y})|) ≫ |(\underline{x} + \underline{y})| <= (|\underline{x}| + |\underline{y}|)⌉, p_{0}, c)]

[splitNumericalSum(−−) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= 0 ⊢ \underline{y} <= 0 ⊢ |(\underline{x} + \underline{y})| <= (|\underline{x}| + |\underline{y}|)] [splitNumericalSum(−−) →^{tex} “splitNumericalSum(--)”]

[splitNumericalSum(−−) →^{pyk} “lemma splitNumericalSum(--)”]

splitNumericalSum(+ − small)

[splitNumericalSum(+ − small) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ \underline{y} <= 0 ⊢ |\underline{y}| <= |\underline{x}| ⊢ LeqAdditionLeft ⊳ \underline{y} <= 0 ≫ (\underline{x} + \underline{y}) <= (\underline{x} + 0); plus0 ≫ (\underline{x} + 0) = \underline{x}; subLeqRight ⊳ (\underline{x} + 0) = \underline{x} ⊳ (\underline{x} + \underline{y}) <= (\underline{x} + 0) ≫ (\underline{x} + \underline{y}) <= \underline{x}; PositiveToRight(Leq)(1term) ⊳ |\underline{y}| <= |\underline{x}| ≫ 0 <= (|\underline{x}| + (−u|\underline{y}|)); lemma nonpositiveNumerical ⊳ \underline{y} <= 0 ≫ |\underline{y}| = (−u\underline{y}); EqNegated ⊳ |\underline{y}| = (−u\underline{y}) ≫ (−u|\underline{y}|) = (−u(−u\underline{y})); DoubleMinus ≫ (−u(−u\underline{y})) = \underline{y}; eqTransitivity ⊳ (−u|\underline{y}|) = (−u(−u\underline{y})) ⊳ (−u(−u\underline{y})) = \underline{y} ≫ (−u|\underline{y}|) = \underline{y}; NonnegativeNumerical ⊳ 0 <= \underline{x} ≫ |\underline{x}| = \underline{x}; AddEquations ⊳ |\underline{x}| = \underline{x} ⊳ (−u|\underline{y}|) = \underline{y} ≫ (|\underline{x}| + (−u|\underline{y}|)) = (\underline{x} + \underline{y}); subLeqRight ⊳ (|\underline{x}| + (−u|\underline{y}|)) = (\underline{x} + \underline{y}) ⊳ 0 <= (|\underline{x}| + (−u|\underline{y}|)) ≫ 0 <= (\underline{x} + \underline{y}); NonnegativeNumerical ⊳ 0 <= (\underline{x} + \underline{y}) ≫ |(\underline{x} + \underline{y})| = (\underline{x} + \underline{y}); eqSymmetry ⊳ |(\underline{x} + \underline{y})| = (\underline{x} + \underline{y}) ≫ (\underline{x} + \underline{y}) = |(\underline{x} + \underline{y})|; eqSymmetry ⊳ |\underline{x}| = \underline{x} ≫ \underline{x} = |\underline{x}|; subLeqLeft ⊳ (\underline{x} + \underline{y}) = |(\underline{x} + \underline{y})| ⊳ (\underline{x} + \underline{y}) <= \underline{x} ≫ |(\underline{x} + \underline{y})| <= \underline{x}; subLeqRight ⊳ \underline{x} = |\underline{x}| ⊳ |(\underline{x} + \underline{y})| <= \underline{x} ≫ |(\underline{x} + \underline{y})| <= |\underline{x}|⌉, p_{0}, c)]

[splitNumericalSum(+ − small) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ \underline{y} <= 0 ⊢ |\underline{y}| <= |\underline{x}| ⊢ |(\underline{x} + \underline{y})| <= |\underline{x}|]

[splitNumericalSum(+ − small) →^{tex} “splitNumericalSum(+-small)”]

[splitNumericalSum(+ − small) →^{pyk} “lemma splitNumericalSum(+-, smallNegative)”]

splitNumericalSum(+ − big)

[splitNumericalSum(+ − big) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ \underline{y} <= 0 ⊢ ˙¬ (|\underline{x}| <= |\underline{y}| ⇒ ˙¬ ( ˙¬ (|\underline{x}| = |\underline{y}|)n)n)n ⊢ NonnegativeNegated ⊳ 0 <= \underline{x} ≫ (−u\underline{x}) <= 0; NonpositiveNegated ⊳ \underline{y} <= 0 ≫ 0 <= (−u\underline{y}); SignNumerical ≫ |\underline{x}| = |(−u\underline{x})|; SubLessLeft ⊳ |\underline{x}| = |(−u\underline{x})| ⊳ ˙¬ (|\underline{x}| <= |\underline{y}| ⇒ ˙¬ ( ˙¬ (|\underline{x}| = |\underline{y}|)n)n)n ≫ ˙¬ (|(−u\underline{x})| <= |\underline{y}| ⇒ ˙¬ ( ˙¬ (|(−u\underline{x})| = |\underline{y}|)n)n)n; SignNumerical ≫ |\underline{y}| = |(−u\underline{y})|; SubLessRight ⊳ |\underline{y}| = |(−u\underline{y})| ⊳ ˙¬ (|(−u\underline{x})| <= |\underline{y}| ⇒ ˙¬ ( ˙¬ (|(−u\underline{x})| = |\underline{y}|)n)n)n ≫ ˙¬ (|(−u\underline{x})| <= |(−u\underline{y})| ⇒ ˙¬ ( ˙¬ (|(−u\underline{x})| = |(−u\underline{y})|)n)n)n; LessLeq ⊳ ˙¬ (|(−u\underline{x})| <= |(−u\underline{y})| ⇒ ˙¬ ( ˙¬ (|(−u\underline{x})| = |(−u\underline{y})|)n)n)n ≫ |(−u\underline{x})| <= |(−u\underline{y})|; splitNumericalSum(+ − small) ⊳ 0 <= (−u\underline{y}) ⊳ (−u\underline{x}) <= 0 ⊳ |(−u\underline{x})| <= |(−u\underline{y})| ≫ |((−u\underline{y}) + (−u\underline{x}))| <= |(−u\underline{y})|; SignNumerical ≫ |(\underline{x} + \underline{y})| = |(−u(\underline{x} + \underline{y}))|; −x − y = −(x + y) ≫ ((−u\underline{x}) + (−u\underline{y})) = (−u(\underline{x} + \underline{y})); plusCommutativity ≫ ((−u\underline{x}) + (−u\underline{y})) = ((−u\underline{y}) + (−u\underline{x})); Equality ⊳ ((−u\underline{x}) + (−u\underline{y})) = (−u(\underline{x} + \underline{y})) ⊳ ((−u\underline{x}) + (−u\underline{y})) = ((−u\underline{y}) + (−u\underline{x})) ≫ (−u(\underline{x} + \underline{y})) = ((−u\underline{y}) + (−u\underline{x})); SameNumerical ⊳ (−u(\underline{x} + \underline{y})) = ((−u\underline{y}) + (−u\underline{x})) ≫ |(−u(\underline{x} + \underline{y}))| = |((−u\underline{y}) + (−u\underline{x}))|; eqTransitivity ⊳ |(\underline{x} + \underline{y})| = |(−u(\underline{x} + \underline{y}))| ⊳ |(−u(\underline{x} + \underline{y}))| = |((−u\underline{y}) + (−u\underline{x}))| ≫ |(\underline{x} + \underline{y})| = |((−u\underline{y}) + (−u\underline{x}))|; eqSymmetry ⊳ |(\underline{x} + \underline{y})| = |((−u\underline{y})+(−u\underline{x}))| ≫ |((−u\underline{y})+(−u\underline{x}))| = |(\underline{x}+\underline{y})|; eqSymmetry⊳|\underline{y}| = |(−u\underline{y})| ≫ |(−u\underline{y})| = |\underline{y}|; subLeqLeft ⊳ |((−u\underline{y}) + (−u\underline{x}))| = |(\underline{x} + \underline{y})| ⊳ |((−u\underline{y}) + (−u\underline{x}))| <= |(−u\underline{y})| ≫ |(\underline{x} + \underline{y})| <= |(−u\underline{y})|; subLeqRight ⊳ |(−u\underline{y})| = |\underline{y}| ⊳ |(\underline{x} + \underline{y})| <= |(−u\underline{y})| ≫ |(\underline{x} + \underline{y})| <= |\underline{y}|⌉, p_{0}, c)]

[splitNumericalSum(+ − big) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ \underline{y} <= 0 ⊢ ˙¬ (|\underline{x}| <= |\underline{y}| ⇒ ˙¬ ( ˙¬ (|\underline{x}| = |\underline{y}|)n)n)n ⊢ |(\underline{x} + \underline{y})| <= |\underline{y}|]

[splitNumericalSum(+ − big) →^{tex} “splitNumericalSum(+-big)”]

[splitNumericalSum(+ − big) →^{pyk} “lemma splitNumericalSum(+-, bigNegative)”]

splitNumericalSum(+−)

[splitNumericalSum(+−) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: |\underline{y}| <= |\underline{x}| ⊢ 0 <= \underline{x} ⊢ \underline{y} <= 0 ⊢ splitNumericalSum(+ − small) ⊳ 0 <= \underline{x} ⊳ \underline{y} <= 0 ⊳ |\underline{y}| <= |\underline{x}| ≫ |(\underline{x} + \underline{y})| <= |\underline{x}|; 0 <= |x| ≫ 0 <= |\underline{y}|; LeqAdditionLeft ⊳ 0 <= |\underline{y}| ≫ (|\underline{x}| + 0) <= (|\underline{x}| + |\underline{y}|); plus0 ≫ (|\underline{x}| + 0) = |\underline{x}|; subLeqLeft ⊳ (|\underline{x}| + 0) = |\underline{x}| ⊳ (|\underline{x}| + 0) <= (|\underline{x}| + |\underline{y}|) ≫ |\underline{x}| <= (|\underline{x}| + |\underline{y}|); leqTransitivity ⊳ |(\underline{x} + \underline{y})| <= |\underline{x}| ⊳ |\underline{x}| <= (|\underline{x}| + |\underline{y}|) ≫ |(\underline{x} + \underline{y})| <= (|\underline{x}| + |\underline{y}|); ∀\underline{x}: ∀\underline{y}: ˙¬ (|\underline{y}| <= |\underline{x}|)n ⊢ 0 <= \underline{x} ⊢ \underline{y} <= 0 ⊢ ToLess ⊳ ˙¬ (|\underline{y}| <= |\underline{x}|)n ≫ ˙¬ (|\underline{x}| <= |\underline{y}| ⇒ ˙¬ ( ˙¬ (|\underline{x}| = |\underline{y}|)n)n)n; splitNumericalSum(+ − big) ⊳ 0 <= \underline{x} ⊳ \underline{y} <= 0 ⊳ ˙¬ (|\underline{x}| <= |\underline{y}| ⇒ ˙¬ ( ˙¬ (|\underline{x}| = |\underline{y}|)n)n)n ≫ |(\underline{x}+\underline{y})| <= |\underline{y}|; 0 <= |x| ≫ 0 <= |\underline{x}|; leqAddition⊳0 <= |\underline{x}| ≫ (0 + |\underline{y}|) <= (|\underline{x}| + |\underline{y}|); plus0Left ≫ (0 + |\underline{y}|) = |\underline{y}|; subLeqLeft ⊳ (0 + |\underline{y}|) = |\underline{y}| ⊳ (0 + |\underline{y}|) <= (|\underline{x}| + |\underline{y}|) ≫ |\underline{y}| <= (|\underline{x}| + |\underline{y}|); leqTransitivity ⊳ |(\underline{x} + \underline{y})| <= |\underline{y}| ⊳ |\underline{y}| <= (|\underline{x}| + |\underline{y}|) ≫ |(\underline{x} + \underline{y})| <= (|\underline{x}| + |\underline{y}|); ∀\underline{x}: ∀\underline{y}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: |\underline{y}| <= |\underline{x}| ⊢ 0 <= \underline{x} ⊢ \underline{y} <= 0 ⊢ |(\underline{x} + \underline{y})| <= (|\underline{x}| + |\underline{y}|) ≫ |\underline{y}| <= |\underline{x}| ⇒ 0 <= \underline{x} ⇒ \underline{y} <= 0 ⇒ |(\underline{x} + \underline{y})| <= (|\underline{x}| + |\underline{y}|); Ded ⊳ ∀\underline{x}: ∀\underline{y}: ˙¬ (|\underline{y}| <= |\underline{x}|)n ⊢ 0 <= \underline{x} ⊢ \underline{y} <= 0 ⊢ |(\underline{x} + \underline{y})| <= (|\underline{x}| + |\underline{y}|) ≫ ˙¬ (|\underline{y}| <= |\underline{x}|)n ⇒ 0 <= \underline{x} ⇒ \underline{y} <= 0 ⇒ |(\underline{x} + \underline{y})| <= (|\underline{x}| + |\underline{y}|); 0 <= \underline{x} ⊢ \underline{y} <= 0 ⊢ FromNegations ⊳ |\underline{y}| <= |\underline{x}| ⇒ 0 <= \underline{x} ⇒ \underline{y} <= 0 ⇒ |(\underline{x}+\underline{y})| <= (|\underline{x}|+|\underline{y}|)⊳ ˙¬ (|\underline{y}| <= |\underline{x}|)n ⇒ 0 <= \underline{x} ⇒ \underline{y} <= 0 ⇒ |(\underline{x}+\underline{y})| <= (|\underline{x}| + |\underline{y}|) ≫ 0 <= \underline{x} ⇒ \underline{y} <= 0 ⇒ |(\underline{x} + \underline{y})| <= (|\underline{x}| + |\underline{y}|); MP2 ⊳ 0 <= \underline{x} ⇒ \underline{y} <= 0 ⇒ |(\underline{x} + \underline{y})| <= (|\underline{x}| + |\underline{y}|) ⊳ 0 <= \underline{x} ⊳ \underline{y} <= 0 ≫ |(\underline{x} + \underline{y})| <= (|\underline{x}| + |\underline{y}|)⌉, p_{0}, c)]

[splitNumericalSum(+−) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ \underline{y} <= 0 ⊢ |(\underline{x} + \underline{y})| <= (|\underline{x}| + |\underline{y}|)]

[splitNumericalSum(+−) →^{tex} “splitNumericalSum(+-)”]

[splitNumericalSum(+−) →^{pyk} “lemma splitNumericalSum(+-)”]

splitNumericalSum(−+)

[splitNumericalSum(−+) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= 0 ⊢ 0 <= \underline{y} ⊢ NonpositiveNegated ⊳ \underline{x} <= 0 ≫ 0 <= (−u\underline{x}); NonnegativeNegated ⊳ 0 <= \underline{y} ≫ (−u\underline{y}) <= 0; splitNumericalSum(+−) ⊳ 0 <= (−u\underline{x}) ⊳ (−u\underline{y}) <= 0 ≫ |((−u\underline{x}) + (−u\underline{y}))| <= (|(−u\underline{x})| + |(−u\underline{y})|); SplitNumericalSumHelper ⊳ |((−u\underline{x}) + (−u\underline{y}))| <= (|(−u\underline{x})| + |(−u\underline{y})|) ≫ |(\underline{x} + \underline{y})| <= (|\underline{x}| + |\underline{y}|)⌉, p_{0}, c)]

[splitNumericalSum(−+) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= 0 ⊢ 0 <= \underline{y} ⊢ |(\underline{x} + \underline{y})| <= (|\underline{x}| + |\underline{y}|)]

[splitNumericalSum(−+) →^{tex} “splitNumericalSum(-+)”]

[splitNumericalSum(−+) →^{pyk} “lemma splitNumericalSum(-+)”]

splitNumericalSum

[splitNumericalSum →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ 0 <= \underline{y} ⊢ splitNumericalSum(++) ⊳ 0 <= \underline{x} ⊳ 0 <= \underline{y} ≫ |(\underline{x} + \underline{y})| <= (|\underline{x}| + |\underline{y}|); ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ \underline{y} <= 0 ⊢ splitNumericalSum(+−) ⊳ 0 <= \underline{x} ⊳ \underline{y} <= 0 ≫ |(\underline{x} + \underline{y})| <= (|\underline{x}| + |\underline{y}|); ∀\underline{x}: ∀\underline{y}: \underline{x} <= 0 ⊢ 0 <= \underline{y} ⊢ splitNumericalSum(−+) ⊳ \underline{x} <= 0 ⊳ 0 <= \underline{y} ≫ |(\underline{x} + \underline{y})| <= (|\underline{x}| + |\underline{y}|); ∀\underline{x}: ∀\underline{y}: \underline{x} <= 0 ⊢ \underline{y} <= 0 ⊢ splitNumericalSum(−−) ⊳ \underline{x} <= 0 ⊳ \underline{y} <= 0 ≫ |(\underline{x} + \underline{y})| <= (|\underline{x}| + |\underline{y}|); ∀\underline{x}: ∀\underline{y}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ 0 <= \underline{y} ⊢ |(\underline{x} + \underline{y})| <= (|\underline{x}| + |\underline{y}|) ≫ 0 <= \underline{x} ⇒ 0 <= \underline{y} ⇒ |(\underline{x} + \underline{y})| <= (|\underline{x}| + |\underline{y}|); Ded ⊳ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ \underline{y} <= 0 ⊢ |(\underline{x} + \underline{y})| <= (|\underline{x}| + |\underline{y}|) ≫ 0 <= \underline{x} ⇒ \underline{y} <= 0 ⇒ |(\underline{x} + \underline{y})| <= (|\underline{x}| + |\underline{y}|); Ded ⊳ ∀\underline{x}: ∀\underline{y}: \underline{x} <= 0 ⊢ 0 <= \underline{y} ⊢ |(\underline{x} + \underline{y})| <= (|\underline{x}| + |\underline{y}|) ≫ \underline{x} <= 0 ⇒ 0 <= \underline{y} ⇒ |(\underline{x} + \underline{y})| <= (|\underline{x}| + |\underline{y}|); Ded ⊳ ∀\underline{x}: ∀\underline{y}: \underline{x} <= 0 ⊢ \underline{y} <= 0 ⊢ |(\underline{x} + \underline{y})| <= (|\underline{x}| + |\underline{y}|) ≫ \underline{x} <= 0 ⇒ \underline{y} <= 0 ⇒ |(\underline{x} + \underline{y})| <= (|\underline{x}| + |\underline{y}|); FromLeqGeq ⊳ 0 <= \underline{x} ⇒ 0 <= \underline{y} ⇒ |(\underline{x} + \underline{y})| <= (|\underline{x}| + |\underline{y}|) ⊳ \underline{x} <= 0 ⇒ 0 <= \underline{y} ⇒ |(\underline{x} + \underline{y})| <= (|\underline{x}| + |\underline{y}|) ≫ 0 <= \underline{y} ⇒ |(\underline{x} + \underline{y})| <= (|\underline{x}| + |\underline{y}|); FromLeqGeq ⊳ 0 <= \underline{x} ⇒ \underline{y} <= 0 ⇒ |(\underline{x} + \underline{y})| <= (|\underline{x}| + |\underline{y}|) ⊳ \underline{x} <= 0 ⇒ \underline{y} <= 0 ⇒ |(\underline{x} + \underline{y})| <= (|\underline{x}| + |\underline{y}|) ≫ \underline{y} <= 0 ⇒ |(\underline{x} + \underline{y})| <= (|\underline{x}| + |\underline{y}|); FromLeqGeq ⊳ 0 <= \underline{y} ⇒ |(\underline{x} + \underline{y})| <= (|\underline{x}| + |\underline{y}|) ⊳ \underline{y} <= 0 ⇒ |(\underline{x} + \underline{y})| <= (|\underline{x}| + |\underline{y}|) ≫ |(\underline{x} + \underline{y})| <= (|\underline{x}| + |\underline{y}|)⌉, p_{0}, c)]

[splitNumericalSum →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: |(\underline{x} + \underline{y})| <= (|\underline{x}| + |\underline{y}|)]

[splitNumericalSum →^{tex} “splitNumericalSum”]

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

SplitNumericalProduct(++)

[SplitNumericalProduct(++) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ 0 <= \underline{y} ⊢ NonnegativeFactors ⊳ 0 <= \underline{x} ⊳ 0 <= \underline{y} ≫ 0 <= (\underline{x} ∗ \underline{y}); NonnegativeNumerical ⊳ 0 <= (\underline{x} ∗ \underline{y}) ≫ |(\underline{x} ∗ \underline{y})| = (\underline{x}∗\underline{y}); NonnegativeNumerical⊳0 <= \underline{x} ≫ |\underline{x}| = \underline{x}; NonnegativeNumerical⊳0 <= \underline{y} ≫ |\underline{y}| = \underline{y}; MultiplyEquations ⊳ |\underline{x}| = \underline{x} ⊳ |\underline{y}| = \underline{y} ≫ (|\underline{x}| ∗ |\underline{y}|) = (\underline{x} ∗ \underline{y}); eqSymmetry ⊳ (|\underline{x}| ∗ |\underline{y}|) = (\underline{x} ∗ \underline{y}) ≫ (\underline{x} ∗ \underline{y}) = (|\underline{x}| ∗ |\underline{y}|); eqTransitivity ⊳ |(\underline{x} ∗ \underline{y})| = (\underline{x} ∗ \underline{y}) ⊳ (\underline{x} ∗ \underline{y}) = (|\underline{x}| ∗ |\underline{y}|) ≫ |(\underline{x} ∗ \underline{y})| = (|\underline{x}| ∗ |\underline{y}|)⌉, p_{0}, c)]

[SplitNumericalProduct(++) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ 0 <= \underline{y} ⊢ |(\underline{x} ∗ \underline{y})| = (|\underline{x}| ∗ |\underline{y}|)]

[SplitNumericalProduct(++) →^{tex} “SplitNumericalProduct(++)”]

[SplitNumericalProduct(++) →^{pyk} “lemma splitNumericalProduct(++)”]

SplitNumericalProduct(+−)

[SplitNumericalProduct(+−) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ \underline{y} <= 0 ⊢ SignNumerical ≫ |(\underline{x} ∗ \underline{y})| = |(−u(\underline{x} ∗ \underline{y}))|; eqSymmetry ⊳ |(\underline{x} ∗ \underline{y})| = |(−u(\underline{x} ∗ \underline{y}))| ≫ |(−u(\underline{x} ∗ \underline{y}))| = |(\underline{x} ∗ \underline{y})|; PlusTimesMinus ≫ (\underline{x} ∗ (−u\underline{y})) = (−u(\underline{x} ∗ \underline{y})); SameNumerical ⊳ (\underline{x} ∗ (−u\underline{y})) = (−u(\underline{x} ∗ \underline{y})) ≫ |(\underline{x} ∗ (−u\underline{y}))| = |(−u(\underline{x}∗\underline{y}))|; eqTransitivity⊳|(\underline{x}∗(−u\underline{y}))| = |(−u(\underline{x}∗\underline{y}))|⊳|(−u(\underline{x}∗\underline{y}))| = |(\underline{x}∗\underline{y})| ≫ |(\underline{x} ∗ (−u\underline{y}))| = |(\underline{x} ∗ \underline{y})|; SignNumerical ≫ |\underline{y}| = |(−u\underline{y})|; eqSymmetry ⊳ |\underline{y}| = |(−u\underline{y})| ≫ |(−u\underline{y})| = |\underline{y}|; EqMultiplicationLeft ⊳ |(−u\underline{y})| = |\underline{y}| ≫ (|\underline{x}| ∗ |(−u\underline{y})|) = (|\underline{x}| ∗ |\underline{y}|); NonpositiveNegated ⊳ \underline{y} <= 0 ≫ 0 <= (−u\underline{y}); SplitNumericalProduct(++) ⊳ 0 <= \underline{x} ⊳ 0 <= (−u\underline{y}) ≫ |(\underline{x} ∗ (−u\underline{y}))| = (|\underline{x}| ∗ |(−u\underline{y})|); eqTransitivity ⊳ |(\underline{x} ∗ (−u\underline{y}))| = (|\underline{x}| ∗ |(−u\underline{y})|) ⊳ (|\underline{x}| ∗ |(−u\underline{y})|) = (|\underline{x}| ∗ |\underline{y}|) ≫ |(\underline{x} ∗ (−u\underline{y}))| = (|\underline{x}| ∗ |\underline{y}|); Equality ⊳ |(\underline{x} ∗ (−u\underline{y}))| = |(\underline{x} ∗ \underline{y})| ⊳ |(\underline{x} ∗ (−u\underline{y}))| = (|\underline{x}| ∗ |\underline{y}|) ≫ |(\underline{x} ∗ \underline{y})| = (|\underline{x}| ∗ |\underline{y}|)⌉, p_{0}, c)]

[SplitNumericalProduct(+−) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ \underline{y} <= 0 ⊢ |(\underline{x} ∗ \underline{y})| = (|\underline{x}| ∗ |\underline{y}|)]

[SplitNumericalProduct(+−) →^{tex} “SplitNumericalProduct(+-)”]

[SplitNumericalProduct(+−) →^{pyk} “lemma splitNumericalProduct(+-)”]

SplitNumericalProduct

[SplitNumericalProduct →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ 0 <= \underline{y} ⊢ SplitNumericalProduct(++) ⊳ 0 <= \underline{x} ⊳ 0 <= \underline{y} ≫ |(\underline{x} ∗ \underline{y})| = (|\underline{x}| ∗ |\underline{y}|); ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ \underline{y} <= 0 ⊢ SplitNumericalProduct(+−) ⊳ 0 <= \underline{x} ⊳ \underline{y} <= 0 ≫ |(\underline{x} ∗ \underline{y})| = (|\underline{x}| ∗ |\underline{y}|); ∀\underline{x}: ∀\underline{y}: \underline{x} <= 0 ⊢ 0 <= \underline{y} ⊢ SplitNumericalProduct(+−) ⊳ 0 <= \underline{y} ⊳ \underline{x} <= 0 ≫ |(\underline{y} ∗ \underline{x})| = (|\underline{y}| ∗ |\underline{x}|); timesCommutativity ≫ (\underline{x} ∗ \underline{y}) = (\underline{y} ∗ \underline{x}); SameNumerical ⊳ (\underline{x} ∗ \underline{y}) = (\underline{y} ∗ \underline{x}) ≫ |(\underline{x} ∗ \underline{y})| = |(\underline{y} ∗ \underline{x})|; timesCommutativity ≫ (|\underline{y}| ∗ |\underline{x}|) = (|\underline{x}| ∗ |\underline{y}|); eqTransitivity4 ⊳ |(\underline{x} ∗ \underline{y})| = |(\underline{y} ∗ \underline{x})| ⊳ |(\underline{y} ∗ \underline{x})| = (|\underline{y}| ∗ |\underline{x}|) ⊳ (|\underline{y}| ∗ |\underline{x}|) = (|\underline{x}| ∗ |\underline{y}|) ≫ |(\underline{x} ∗ \underline{y})| = (|\underline{x}| ∗ |\underline{y}|); ∀\underline{x}: ∀\underline{y}: \underline{x} <= 0 ⊢ \underline{y} <= 0 ⊢ NonpositiveNegated ⊳ \underline{x} <= 0 ≫ 0 <= (−u\underline{x}); NonpositiveNegated ⊳ \underline{y} <= 0 ≫ 0 <= (−u\underline{y}); SplitNumericalProduct(++) ⊳ 0 <= (−u\underline{x}) ⊳ 0 <= (−u\underline{y}) ≫ |((−u\underline{x}) ∗ (−u\underline{y}))| = (|(−u\underline{x})| ∗ |(−u\underline{y})|); MinusTimesMinus ≫ ((−u\underline{x}) ∗ (−u\underline{y})) = (\underline{x} ∗ \underline{y}); SameNumerical ⊳ ((−u\underline{x}) ∗ (−u\underline{y})) = (\underline{x} ∗ \underline{y}) ≫ |((−u\underline{x}) ∗ (−u\underline{y}))| = |(\underline{x} ∗ \underline{y})|; eqSymmetry ⊳ |((−u\underline{x}) ∗ (−u\underline{y}))| = |(\underline{x} ∗ \underline{y})| ≫ |(\underline{x} ∗ \underline{y})| = |((−u\underline{x}) ∗ (−u\underline{y}))|; SignNumerical ≫ |\underline{x}| = |(−u\underline{x})|; SignNumerical ≫ |\underline{y}| = |(−u\underline{y})|; MultiplyEquations ⊳ |\underline{x}| = |(−u\underline{x})| ⊳ |\underline{y}| = |(−u\underline{y})| ≫ (|\underline{x}| ∗ |\underline{y}|) = (|(−u\underline{x})| ∗ |(−u\underline{y})|); eqSymmetry ⊳ (|\underline{x}| ∗ |\underline{y}|) = (|(−u\underline{x})| ∗ |(−u\underline{y})|) ≫ (|(−u\underline{x})| ∗ |(−u\underline{y})|) = (|\underline{x}| ∗ |\underline{y}|); eqTransitivity4 ⊳ |(\underline{x} ∗ \underline{y})| = |((−u\underline{x})∗(−u\underline{y}))|⊳|((−u\underline{x})∗(−u\underline{y}))| = (|(−u\underline{x})|∗|(−u\underline{y})|)⊳(|(−u\underline{x})|∗|(−u\underline{y})|) = (|\underline{x}|∗|\underline{y}|) ≫ |(\underline{x}∗\underline{y})| = (|\underline{x}|∗|\underline{y}|); ∀\underline{x}: ∀\underline{y}: Ded⊳∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ 0 <= \underline{y} ⊢ |(\underline{x}∗\underline{y})| = (|\underline{x}| ∗ |\underline{y}|) ≫ 0 <= \underline{x} ⇒ 0 <= \underline{y} ⇒ |(\underline{x} ∗ \underline{y})| = (|\underline{x}| ∗ |\underline{y}|); Ded ⊳ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ \underline{y} <= 0 ⊢ |(\underline{x} ∗ \underline{y})| = (|\underline{x}| ∗ |\underline{y}|) ≫ 0 <= \underline{x} ⇒ \underline{y} <= 0 ⇒ |(\underline{x} ∗ \underline{y})| = (|\underline{x}| ∗ |\underline{y}|); Ded ⊳ ∀\underline{x}: ∀\underline{y}: \underline{x} <= 0 ⊢ 0 <= \underline{y} ⊢ |(\underline{x} ∗ \underline{y})| = (|\underline{x}| ∗ |\underline{y}|) ≫ \underline{x} <= 0 ⇒ 0 <= \underline{y} ⇒ |(\underline{x} ∗ \underline{y})| = (|\underline{x}| ∗ |\underline{y}|); Ded ⊳ ∀\underline{x}: ∀\underline{y}: \underline{x} <= 0 ⊢ \underline{y} <= 0 ⊢ |(\underline{x} ∗ \underline{y})| = (|\underline{x}| ∗ |\underline{y}|) ≫ \underline{x} <= 0 ⇒ \underline{y} <= 0 ⇒ |(\underline{x} ∗ \underline{y})| = (|\underline{x}| ∗ |\underline{y}|); FromLeqGeq ⊳ 0 <= \underline{x} ⇒ 0 <= \underline{y} ⇒ |(\underline{x} ∗ \underline{y})| = (|\underline{x}| ∗ |\underline{y}|) ⊳ \underline{x} <= 0 ⇒ 0 <= \underline{y} ⇒ |(\underline{x} ∗ \underline{y})| = (|\underline{x}| ∗ |\underline{y}|) ≫ 0 <= \underline{y} ⇒ |(\underline{x} ∗ \underline{y})| = (|\underline{x}| ∗ |\underline{y}|); FromLeqGeq ⊳ 0 <= \underline{x} ⇒ \underline{y} <= 0 ⇒ |(\underline{x} ∗ \underline{y})| = (|\underline{x}| ∗ |\underline{y}|) ⊳ \underline{x} <= 0 ⇒ \underline{y} <= 0 ⇒ |(\underline{x} ∗ \underline{y})| = (|\underline{x}| ∗ |\underline{y}|) ≫ \underline{y} <= 0 ⇒ |(\underline{x} ∗ \underline{y})| = (|\underline{x}| ∗ |\underline{y}|); FromLeqGeq ⊳ 0 <= \underline{y} ⇒ |(\underline{x} ∗ \underline{y})| = (|\underline{x}| ∗ |\underline{y}|) ⊳ \underline{y} <= 0 ⇒ |(\underline{x} ∗ \underline{y})| = (|\underline{x}| ∗ |\underline{y}|) ≫ |(\underline{x} ∗ \underline{y})| = (|\underline{x}| ∗ |\underline{y}|)⌉, p_{0}, c)]

[SplitNumericalProduct →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: |(\underline{x} ∗ \underline{y})| = (|\underline{x}| ∗ |\underline{y}|)]

[SplitNumericalProduct →^{tex} “SplitNumericalProduct”]

[SplitNumericalProduct →^{pyk} “lemma splitNumericalProduct”]

insertMiddleTerm(Numerical)

[insertMiddleTerm(Numerical) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: splitNumericalSum ≫ |((\underline{x} + (−u\underline{z})) + (\underline{z} + \underline{y}))| <= (|(\underline{x} + (−u\underline{z}))| + |(\underline{z} + \underline{y})|); insertMiddleTerm(Sum) ≫ (\underline{x} + \underline{y}) = ((\underline{x} + (−u\underline{z})) + (\underline{z} + \underline{y})); SameNumerical ⊳ (\underline{x} + \underline{y}) = ((\underline{x} + (−u\underline{z})) + (\underline{z} + \underline{y})) ≫ |(\underline{x} + \underline{y})| = |((\underline{x} + (−u\underline{z})) + (\underline{z} + \underline{y}))|; eqSymmetry ⊳ |(\underline{x} + \underline{y})| = |((\underline{x} + (−u\underline{z})) + (\underline{z} + \underline{y}))| ≫ |((\underline{x} + (−u\underline{z})) + (\underline{z} + \underline{y}))| = |(\underline{x}+\underline{y})|; subLeqLeft⊳|((\underline{x}+(−u\underline{z}))+(\underline{z}+\underline{y}))| = |(\underline{x}+\underline{y})|⊳|((\underline{x}+(−u\underline{z}))+(\underline{z}+\underline{y}))| <= (|(\underline{x} + (−u\underline{z}))| + |(\underline{z} + \underline{y})|) ≫ |(\underline{x} + \underline{y})| <= (|(\underline{x} + (−u\underline{z}))| + |(\underline{z} + \underline{y})|)⌉, p_{0}, c)] [insertMiddleTerm(Numerical) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: |(\underline{x} + \underline{y})| <= (|(\underline{x} + (−u\underline{z}))| + |(\underline{z} + \underline{y})|)]

[insertMiddleTerm(Numerical) →^{tex} “insertMiddleTerm(Numerical)”]

[insertMiddleTerm(Numerical) →^{pyk} “lemma insertMiddleTerm(Numerical)”]

insertTwoMiddleTerms(Numerical)

[insertTwoMiddleTerms(Numerical) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: insertMiddleTerm(Numerical) ≫ |(\underline{x} + \underline{y})| <= (|(\underline{x} + (−u\underline{z}))| + |(\underline{z} + \underline{y})|); insertMiddleTerm(Numerical) ≫ |(\underline{z} + \underline{y})| <= (|(\underline{z} + (−u\underline{u}))| + |(\underline{u} + \underline{y})|); LeqAdditionLeft ⊳ |(\underline{z} + \underline{y})| <= (|(\underline{z} + (−u\underline{u}))| + |(\underline{u} + \underline{y})|) ≫ (|(\underline{x} + (−u\underline{z}))| + |(\underline{z} + \underline{y})|) <= (|(\underline{x} + (−u\underline{z}))| + (|(\underline{z} + (−u\underline{u}))| + |(\underline{u} + \underline{y})|)); leqTransitivity ⊳ |(\underline{x} + \underline{y})| <= (|(\underline{x} + (−u\underline{z}))| + |(\underline{z} + \underline{y})|) ⊳ (|(\underline{x} + (−u\underline{z}))| + |(\underline{z} + \underline{y})|) <= (|(\underline{x} + (−u\underline{z}))| + (|(\underline{z} + (−u\underline{u}))| + |(\underline{u} + \underline{y})|)) ≫ |(\underline{x} + \underline{y})| <= (|(\underline{x} + (−u\underline{z}))| + (|(\underline{z} + (−u\underline{u}))| + |(\underline{u} + \underline{y})|)); plusAssociativity ≫ ((|(\underline{x}+(−u\underline{z}))|+|(\underline{z}+(−u\underline{u}))|)+|(\underline{u}+\underline{y})|) = (|(\underline{x}+(−u\underline{z}))|+(|(\underline{z}+(−u\underline{u}))|+|(\underline{u}+ \underline{y})|)); eqSymmetry ⊳((|(\underline{x}+(−u\underline{z}))|+|(\underline{z}+(−u\underline{u}))|)+|(\underline{u}+\underline{y})|) = (|(\underline{x}+(−u\underline{z}))|+ (|(\underline{z} + (−u\underline{u}))| + |(\underline{u} + \underline{y})|)) ≫ (|(\underline{x} + (−u\underline{z}))| + (|(\underline{z} + (−u\underline{u}))| + |(\underline{u} + \underline{y})|)) = ((|(\underline{x} + (−u\underline{z}))| + |(\underline{z} + (−u\underline{u}))|) + |(\underline{u} + \underline{y})|); subLeqRight ⊳ (|(\underline{x} + (−u\underline{z}))| + (|(\underline{z} + (−u\underline{u}))| + |(\underline{u} + \underline{y})|)) = ((|(\underline{x} + (−u\underline{z}))| + |(\underline{z} + (−u\underline{u}))|) + |(\underline{u} + \underline{y})|) ⊳ |(\underline{x} + \underline{y})| <= (|(\underline{x} + (−u\underline{z}))| + (|(\underline{z} + (−u\underline{u}))| + |(\underline{u} + \underline{y})|)) ≫ |(\underline{x} + \underline{y})| <= ((|(\underline{x} + (−u\underline{z}))| + |(\underline{z} + (−u\underline{u}))|) + |(\underline{u} + \underline{y})|)⌉, p_{0}, c)]

[insertTwoMiddleTerms(Numerical) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: |(\underline{x} + \underline{y})| <= ((|(\underline{x} + (−u\underline{z}))| + |(\underline{z} + (−u\underline{u}))|) + |(\underline{u} + \underline{y})|)]

[insertTwoMiddleTerms(Numerical) →^{tex} “insertTwoMiddleTerms(Numerical)”]

[insertTwoMiddleTerms(Numerical) →^{pyk} “lemma insertTwoMiddleTerms(Numerical)”]

Three2twoTerms

[Three2twoTerms →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: (\underline{y} + \underline{z}) = \underline{u} ⊢ EqAdditionLeft ⊳ (\underline{y} + \underline{z}) = \underline{u} ≫ (\underline{x} + (\underline{y} + \underline{z})) = (\underline{x} + \underline{u}); plusAssociativity ≫ ((\underline{x} + \underline{y}) + \underline{z}) = (\underline{x} + (\underline{y} + \underline{z})); eqTransitivity ⊳ ((\underline{x} + \underline{y}) + \underline{z}) = (\underline{x} + (\underline{y} + \underline{z})) ⊳ (\underline{x} + (\underline{y} + \underline{z})) = (\underline{x} + \underline{u}) ≫ ((\underline{x} + \underline{y}) + \underline{z}) = (\underline{x} + \underline{u})⌉, p_{0}, c)]

[Three2twoTerms →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: (\underline{y} + \underline{z}) = \underline{u} ⊢ ((\underline{x} + \underline{y}) + \underline{z}) = (\underline{x} + \underline{u})]

[Three2twoTerms →^{tex} “Three2twoTerms”] [Three2twoTerms →^{pyk} “lemma three2twoTerms”]

Three2threeTerms

[Three2threeTerms →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: plusCommutativity ≫ (\underline{y} + \underline{z}) = (\underline{z} + \underline{y}); Three2twoTerms ⊳ (\underline{y} + \underline{z}) = (\underline{z} + \underline{y}) ≫ ((\underline{x} + \underline{y}) + \underline{z}) = (\underline{x} + (\underline{z} + \underline{y})); plusAssociativity ≫ ((\underline{x}+\underline{z})+\underline{y}) = (\underline{x}+(\underline{z}+\underline{y})); eqSymmetry⊳((\underline{x}+\underline{z})+\underline{y}) = (\underline{x} + (\underline{z} + \underline{y})) ≫ (\underline{x} + (\underline{z} + \underline{y})) = ((\underline{x} + \underline{z}) + \underline{y}); eqTransitivity ⊳ ((\underline{x} + \underline{y}) + \underline{z}) = (\underline{x} + (\underline{z} + \underline{y})) ⊳ (\underline{x} + (\underline{z} + \underline{y})) = ((\underline{x} + \underline{z}) + \underline{y}) ≫ ((\underline{x} + \underline{y}) + \underline{z}) = ((\underline{x} + \underline{z}) + \underline{y})⌉, p_{0}, c)]

[Three2threeTerms →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ((\underline{x} + \underline{y}) + \underline{z}) = ((\underline{x} + \underline{z}) + \underline{y})]

[Three2threeTerms →^{tex} “Three2threeTerms”]

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

Three2twoFactors

[Three2twoFactors →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: (\underline{y} ∗ \underline{z}) = \underline{u} ⊢ EqMultiplicationLeft ⊳ (\underline{y} ∗ \underline{z}) = \underline{u} ≫ (\underline{x} ∗ (\underline{y} ∗ \underline{z})) = (\underline{x} ∗ \underline{u}); timesAssociativity ≫ ((\underline{x} ∗ \underline{y}) ∗ \underline{z}) = (\underline{x} ∗ (\underline{y} ∗ \underline{z})); eqTransitivity ⊳ ((\underline{x} ∗ \underline{y}) ∗ \underline{z}) = (\underline{x} ∗ (\underline{y} ∗ \underline{z})) ⊳ (\underline{x} ∗ (\underline{y} ∗ \underline{z})) = (\underline{x} ∗ \underline{u}) ≫ ((\underline{x} ∗ \underline{y}) ∗ \underline{z}) = (\underline{x} ∗ \underline{u})⌉, p_{0}, c)]

[Three2twoFactors →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: (\underline{y}∗\underline{z}) = \underline{u} ⊢ ((\underline{x}∗\underline{y})∗\underline{z}) = (\underline{x}∗\underline{u})]

[Three2twoFactors →^{tex} “Three2twoFactors”]

[Three2twoFactors →^{pyk} “lemma three2twoFactors”]

Three2threeFactors

[Three2threeFactors →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: timesCommutativity ≫ (\underline{y} ∗ \underline{z}) = (\underline{z} ∗ \underline{y}); Three2twoFactors ⊳ (\underline{y} ∗ \underline{z}) = (\underline{z} ∗ \underline{y}) ≫ ((\underline{x} ∗ \underline{y}) ∗ \underline{z}) = (\underline{x} ∗ (\underline{z} ∗ \underline{y})); timesAssociativity ≫ ((\underline{x} ∗ \underline{z}) ∗ \underline{y}) = (\underline{x} ∗ (\underline{z} ∗ \underline{y})); eqSymmetry ⊳ ((\underline{x} ∗ \underline{z}) ∗ \underline{y}) = (\underline{x} ∗ (\underline{z} ∗ \underline{y})) ≫ (\underline{x} ∗ (\underline{z} ∗ \underline{y})) = ((\underline{x} ∗ \underline{z}) ∗ \underline{y}); eqTransitivity ⊳ ((\underline{x} ∗ \underline{y}) ∗ \underline{z}) = (\underline{x} ∗ (\underline{z} ∗ \underline{y})) ⊳ (\underline{x} ∗ (\underline{z} ∗ \underline{y})) = ((\underline{x} ∗ \underline{z}) ∗ \underline{y}) ≫ ((\underline{x} ∗ \underline{y}) ∗ \underline{z}) = ((\underline{x} ∗ \underline{z}) ∗ \underline{y})⌉, p_{0}, c)]

[Three2threeFactors →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ((\underline{x} ∗ \underline{y}) ∗ \underline{z}) = ((\underline{x} ∗ \underline{z}) ∗ \underline{y})]

[Three2threeFactors →^{tex} “Three2threeFactors”]

[Three2threeFactors →^{pyk} “lemma three2threeFactors”]

Times(−1)

[Times(−1) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: Negative ≫ (1 + (−u1)) = 0; plusCommutativity ≫ ((−u1) + 1) = (1 + (−u1)); eqTransitivity ⊳ ((−u1) + 1) = (1 + (−u1)) ⊳ (1 + (−u1)) = 0 ≫ ((−u1) + 1) = 0; EqMultiplicationLeft ⊳ ((−u1) + 1) = 0 ≫ (\underline{x} ∗ ((−u1) + 1)) = (\underline{x} ∗ 0); x ∗ 0 = 0 ≫ (\underline{x} ∗ 0) = 0; eqTransitivity ⊳ (\underline{x} ∗ ((−u1) + 1)) = (\underline{x} ∗ 0) ⊳ (\underline{x} ∗ 0) = 0 ≫ (\underline{x} ∗ ((−u1) + 1)) = 0; Distribution ≫ (\underline{x} ∗ ((−u1) + 1)) = ((\underline{x} ∗ (−u1)) + (\underline{x} ∗ 1)); eqSymmetry ⊳ (\underline{x} ∗ ((−u1) + 1)) = ((\underline{x} ∗ (−u1)) + (\underline{x} ∗ 1)) ≫ ((\underline{x}∗(−u1))+(\underline{x}∗1)) = (\underline{x}∗((−u1)+1)); eqTransitivity⊳((\underline{x}∗(−u1))+(\underline{x}∗1)) = (\underline{x} ∗ ((−u1) + 1)) ⊳ (\underline{x} ∗ ((−u1) + 1)) = 0 ≫ ((\underline{x} ∗ (−u1)) + (\underline{x} ∗ 1)) = 0; PositiveToRight(Eq) ⊳ ((\underline{x} ∗ (−u1)) + (\underline{x} ∗ 1)) = 0 ≫ (\underline{x} ∗ (−u1)) = (0 + (−u(\underline{x} ∗ 1))); plus0Left ≫ (0 + (−u(\underline{x} ∗ 1))) = (−u(\underline{x} ∗ 1)); eqTransitivity ⊳ (\underline{x} ∗ (−u1)) = (0 + (−u(\underline{x} ∗ 1))) ⊳ (0 + (−u(\underline{x} ∗ 1))) = (−u(\underline{x} ∗ 1)) ≫ (\underline{x} ∗ (−u1)) = (−u(\underline{x} ∗ 1)); times1 ≫ (\underline{x} ∗ 1) = \underline{x}; EqNegated ⊳ (\underline{x} ∗ 1) = \underline{x} ≫ (−u(\underline{x} ∗ 1)) = (−u\underline{x}); eqTransitivity ⊳ (\underline{x} ∗ (−u1)) = (−u(\underline{x} ∗ 1)) ⊳ (−u(\underline{x} ∗ 1)) = (−u\underline{x}) ≫ (\underline{x} ∗ (−u1)) = (−u\underline{x})⌉, p_{0}, c)]

[Times(−1) →^{stmt} SystemQ ⊢ ∀\underline{x}: (\underline{x} ∗ (−u1)) = (−u\underline{x})]

[Times(−1) →^{tex} “Times(-1)”]

[Times(−1) →^{pyk} “lemma times(-1)”]

Times(−1)Left

[Times(−1)Left →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: Times(−1) ≫ (\underline{x} ∗ (−u1)) = (−u\underline{x}); timesCommutativity ≫ ((−u1) ∗ \underline{x}) = (\underline{x} ∗ (−u1)); eqTransitivity ⊳ ((−u1) ∗ \underline{x}) = (\underline{x} ∗ (−u1)) ⊳ (\underline{x} ∗ (−u1)) = (−u\underline{x}) ≫ ((−u1) ∗ \underline{x}) = (−u\underline{x})⌉, p_{0}, c)]

[Times(−1)Left →^{stmt} SystemQ ⊢ ∀\underline{x}: ((−u1) ∗ \underline{x}) = (−u\underline{x})]

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

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

MaxLeq(1)

[MaxLeq(1) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: \underline{y} <= \underline{x} ⊢ FromMax(1) ⊳ \underline{y} <= \underline{x} ≫ if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) = \underline{x}; eqSymmetry ⊳ if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) = \underline{x} ≫ \underline{x} = if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}); eqLeq ⊳ \underline{x} = if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) ≫ \underline{x} <= if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}); ∀\underline{x}: ∀\underline{y}: ˙¬ (\underline{y} <= \underline{x})n ⊢ FromMax(2) ⊳ ˙¬ (\underline{y} <= \underline{x})n ≫ if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) = \underline{y}; eqSymmetry ⊳ if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) = \underline{y} ≫ \underline{y} = if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}); ToLess ⊳ ˙¬ (\underline{y} <= \underline{x})n ≫ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n; LessLeq ⊳ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ≫ \underline{x} <= \underline{y}; subLeqRight ⊳ \underline{y} = if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) ⊳ \underline{x} <= \underline{y} ≫ \underline{x} <= if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}); ∀\underline{x}: ∀\underline{y}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: \underline{y} <= \underline{x} ⊢ \underline{x} <= if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) ≫ \underline{y} <= \underline{x} ⇒ \underline{x} <= if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}); Ded ⊳ ∀\underline{x}: ∀\underline{y}: ˙¬ (\underline{y} <= \underline{x})n ⊢ \underline{x} <= if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) ≫ ˙¬ (\underline{y} <= \underline{x})n ⇒ \underline{x} <= if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}); FromNegations ⊳ \underline{y} <= \underline{x} ⇒ \underline{x} <= if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) ⊳ ˙¬ (\underline{y} <= \underline{x})n ⇒ \underline{x} <= if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) ≫ \underline{x} <= if(\underline{y} <= \underline{x}, \underline{x}, \underline{y})⌉, p_{0}, c)]

[MaxLeq(1) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: \underline{x} <= if(\underline{y} <= \underline{x}, \underline{x}, \underline{y})]

[MaxLeq(1) →^{tex} “MaxLeq(1)”]

[MaxLeq(1) →^{pyk} “lemma leqMax1”]

MaxLeq(2)

[MaxLeq(2) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: \underline{y} <= \underline{x} ⊢ FromMax(1) ⊳ \underline{y} <= \underline{x} ≫ if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) = \underline{x}; eqSymmetry ⊳ if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) = \underline{x} ≫ \underline{x} = if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}); subLeqRight ⊳ \underline{x} = if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) ⊳ \underline{y} <= \underline{x} ≫ \underline{y} <= if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}); ∀\underline{x}: ∀\underline{y}: ˙¬ (\underline{y} <= \underline{x})n ⊢ FromMax(2) ⊳ ˙¬ (\underline{y} <= \underline{x})n ≫ if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) = \underline{y}; eqSymmetry ⊳ if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) = \underline{y} ≫ \underline{y} = if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}); eqLeq ⊳ \underline{y} = if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) ≫ \underline{y} <= if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}); ∀\underline{x}: ∀\underline{y}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: \underline{y} <= \underline{x} ⊢ \underline{y} <= if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) ≫ \underline{y} <= \underline{x} ⇒ \underline{y} <= if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}); Ded ⊳ ∀\underline{x}: ∀\underline{y}: ˙¬ (\underline{y} <= \underline{x})n ⊢ \underline{y} <= if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) ≫ ˙¬ (\underline{y} <= \underline{x})n ⇒ \underline{y} <= if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}); FromNegations ⊳ \underline{y} <= \underline{x} ⇒ \underline{y} <= if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) ⊳ ˙¬ (\underline{y} <= \underline{x})n ⇒ \underline{y} <= if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) ≫ \underline{y} <= if(\underline{y} <= \underline{x}, \underline{x}, \underline{y})⌉, p_{0}, c)]

[MaxLeq(2) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: \underline{y} <= if(\underline{y} <= \underline{x}, \underline{x}, \underline{y})]

[MaxLeq(2) →^{tex} “MaxLeq(2)”]

[MaxLeq(2) →^{pyk} “lemma leqMax2”]

LessThanMax

[LessThanMax →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{a} ⇒ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⊢ \underline{a} ⊢ MP ⊳ \underline{a} ⇒ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⊳ \underline{a} ≫ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n; MaxLeq(1) ≫ \underline{y} <= if(\underline{z} <= \underline{y}, \underline{y}, \underline{z}); LessLeqTransitivity ⊳ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⊳ \underline{y} <= if(\underline{z} <= \underline{y}, \underline{y}, \underline{z}) ≫ ˙¬ (\underline{x} <= if(\underline{z} <= \underline{y}, \underline{y}, \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = if(\underline{z} <= \underline{y}, \underline{y}, \underline{z}))n)n)n; ∀\underline{a}: ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ (\underline{a})n ⇒ ˙¬ (\underline{x} <= \underline{z} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{z})n)n)n ⊢ ˙¬ (\underline{a})n ⊢ MP ⊳ ˙¬ (\underline{a})n ⇒ ˙¬ (\underline{x} <= \underline{z} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{z})n)n)n ⊳ ˙¬ (\underline{a})n ≫ ˙¬ (\underline{x} <= \underline{z} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{z})n)n)n; MaxLeq(2) ≫ \underline{z} <= if(\underline{z} <= \underline{y}, \underline{y}, \underline{z}); LessLeqTransitivity ⊳ ˙¬ (\underline{x} <= \underline{z} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{z})n)n)n ⊳ \underline{z} <= if(\underline{z} <= \underline{y}, \underline{y}, \underline{z}) ≫ ˙¬ (\underline{x} <= if(\underline{z} <= \underline{y}, \underline{y}, \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = if(\underline{z} <= \underline{y}, \underline{y}, \underline{z}))n)n)n; ∀\underline{a}: ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: Ded ⊳ ∀\underline{a}: ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{a} ⇒ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⊢ \underline{a} ⊢ ˙¬ (\underline{x} <= if(\underline{z} <= \underline{y}, \underline{y}, \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = if(\underline{z} <= \underline{y}, \underline{y}, \underline{z}))n)n)n ≫ \underline{a} ⇒ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⇒ \underline{a} ⇒ ˙¬ (\underline{x} <= if(\underline{z} <= \underline{y}, \underline{y}, \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = if(\underline{z} <= \underline{y}, \underline{y}, \underline{z}))n)n)n; Ded ⊳ ∀\underline{a}: ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ (\underline{a})n ⇒ ˙¬ (\underline{x} <= \underline{z} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{z})n)n)n ⊢ ˙¬ (\underline{a})n ⊢ ˙¬ (\underline{x} <= if(\underline{z} <= \underline{y}, \underline{y}, \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = if(\underline{z} <= \underline{y}, \underline{y}, \underline{z}))n)n)n ≫ ˙¬ (\underline{a})n ⇒ ˙¬ (\underline{x} <= \underline{z} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{z})n)n)n ⇒ ˙¬ (\underline{a})n ⇒ ˙¬ (\underline{x} <= if(\underline{z} <= \underline{y}, \underline{y}, \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = if(\underline{z} <= \underline{y}, \underline{y}, \underline{z}))n)n)n; \underline{a} ⇒ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⊢ ˙¬ (\underline{a})n ⇒ ˙¬ (\underline{x} <= \underline{z} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{z})n)n)n ⊢ MP ⊳ \underline{a} ⇒ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⇒ \underline{a} ⇒ ˙¬ (\underline{x} <= if(\underline{z} <= \underline{y}, \underline{y}, \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = if(\underline{z} <= \underline{y}, \underline{y}, \underline{z}))n)n)n ⊳ \underline{a} ⇒ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ≫ \underline{a} ⇒ ˙¬ (\underline{x} <= if(\underline{z} <= \underline{y}, \underline{y}, \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = if(\underline{z} <= \underline{y}, \underline{y}, \underline{z}))n)n)n; MP ⊳ ˙¬ (\underline{a})n ⇒ ˙¬ (\underline{x} <= \underline{z} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{z})n)n)n ⇒ ˙¬ (\underline{a})n ⇒ ˙¬ (\underline{x} <= if(\underline{z} <= \underline{y}, \underline{y}, \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = if(\underline{z} <= \underline{y}, \underline{y}, \underline{z}))n)n)n ⊳ ˙¬ (\underline{a})n ⇒ ˙¬ (\underline{x} <= \underline{z} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{z})n)n)n ≫ ˙¬ (\underline{a})n ⇒ ˙¬ (\underline{x} <= if(\underline{z} <= \underline{y}, \underline{y}, \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = if(\underline{z} <= \underline{y}, \underline{y}, \underline{z}))n)n)n; FromNegations ⊳ \underline{a} ⇒ ˙¬ (\underline{x} <= if(\underline{z} <= \underline{y}, \underline{y}, \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = if(\underline{z} <= \underline{y}, \underline{y}, \underline{z}))n)n)n ⊳ ˙¬ (\underline{a})n ⇒ ˙¬ (\underline{x} <= if(\underline{z} <= \underline{y}, \underline{y}, \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = if(\underline{z} <= \underline{y}, \underline{y}, \underline{z}))n)n)n ≫ ˙¬ (\underline{x} <= if(\underline{z} <= \underline{y}, \underline{y}, \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = if(\underline{z} <= \underline{y}, \underline{y}, \underline{z}))n)n)n⌉, p_{0}, c)]

[LessThanMax →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{a} ⇒ ˙¬ (\underline{x} <= \underline{y} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{y})n)n)n ⊢ ˙¬ (\underline{a})n ⇒ ˙¬ (\underline{x} <= \underline{z} ⇒ ˙¬ ( ˙¬ (\underline{x} = \underline{z})n)n)n ⊢ ˙¬ (\underline{x} <= if(\underline{z} <= \underline{y}, \underline{y}, \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = if(\underline{z} <= \underline{y}, \underline{y}, \underline{z}))n)n)n]

[LessThanMax →^{tex} “LessThanMax”]

[LessThanMax →^{pyk} “lemma lessThanMax”]

x + y = zBackwards

[x + y = zBackwards →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: (\underline{x} + \underline{y}) = \underline{z} ⊢ plusCommutativity ≫ (\underline{x} + \underline{y}) = (\underline{y} + \underline{x}); Equality ⊳ (\underline{x} + \underline{y}) = \underline{z} ≫ \underline{z} = (\underline{y} + \underline{x})⌉, p_{0}, c)]

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

[x + y = zBackwards →^{tex} “x+y=zBackwards”]

[x + y = zBackwards →^{pyk} “lemma x+y=zBackwards”]

x ∗ y = zBackwards

[x ∗ y = zBackwards →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: (\underline{x} ∗ \underline{y}) = \underline{z} ⊢ timesCommutativity ≫ (\underline{x}∗\underline{y}) = (\underline{y}∗\underline{x}); Equality⊳(\underline{x}∗\underline{y}) = \underline{z} ≫ \underline{z} = (\underline{y}∗\underline{x})⌉, p_{0}, c)]

[x ∗ y = zBackwards →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: (\underline{x} ∗ \underline{y}) = \underline{z} ⊢ \underline{z} = (\underline{y} ∗ \underline{x})]

[x ∗ y = zBackwards →^{tex} “x∗y=zBackwards”]

[x ∗ y = zBackwards →^{pyk} “lemma x∗y=zBackwards”]

x = x + (y − y)

[x = x + (y − y) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: plus0 ≫ (\underline{x} + 0) = \underline{x}; Negative ≫ (\underline{y} + (−u\underline{y})) = 0; eqSymmetry ⊳ (\underline{y} + (−u\underline{y})) = 0 ≫ 0 = (\underline{y} + (−u\underline{y})); EqAdditionLeft ⊳ 0 = (\underline{y} + (−u\underline{y})) ≫ (\underline{x} + 0) = (\underline{x} + (\underline{y} + (−u\underline{y}))); Equality ⊳ (\underline{x} + 0) = \underline{x} ⊳ (\underline{x} + 0) = (\underline{x} + (\underline{y} + (−u\underline{y}))) ≫ \underline{x} = (\underline{x} + (\underline{y} + (−u\underline{y})))⌉, p_{0}, c)]

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

[x = x + (y − y) →^{tex} “x=x+(y-y)”]

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

x = x + y − y

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

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

[x = x + y − y →^{tex} “x=x+y-y”]

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

x = x ∗ y ∗ (1/y)

[x = x ∗ y ∗ (1/y) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ (\underline{y} = 0)n ⊢ times1 ≫ (\underline{x} ∗ 1) = \underline{x}; Reciprocal⊳ ˙¬ (\underline{y} = 0)n ≫ (\underline{y} ∗rec\underline{y}) = 1; Three2twoFactors⊳(\underline{y} ∗rec\underline{y}) = 1 ≫ ((\underline{x}∗\underline{y})∗rec\underline{y}) = (\underline{x}∗1); eqTransitivity⊳((\underline{x}∗\underline{y})∗rec\underline{y}) = (\underline{x}∗1)⊳(\underline{x}∗1) = \underline{x} ≫ ((\underline{x}∗\underline{y})∗rec\underline{y}) = \underline{x}; eqSymmetry ⊳((\underline{x}∗\underline{y})∗rec\underline{y}) = \underline{x} ≫ \underline{x} = ((\underline{x}∗\underline{y})∗rec\underline{y})⌉, p_{0}, c)]

[x = x ∗ y ∗ (1/y) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ (\underline{y} = 0)n ⊢ \underline{x} = ((\underline{x} ∗ \underline{y}) ∗ rec\underline{y})]

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

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

insertMiddleTerm(Sum)

[insertMiddleTerm(Sum) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: x = x + y − y ≫ \underline{x} = ((\underline{x} + \underline{z}) + (−u\underline{z})); Three2threeTerms ≫ ((\underline{x} + \underline{z}) + (−u\underline{z})) = ((\underline{x} + (−u\underline{z})) + \underline{z}); eqTransitivity ⊳ \underline{x} = ((\underline{x} + \underline{z}) + (−u\underline{z})) ⊳ ((\underline{x} + \underline{z}) + (−u\underline{z})) = ((\underline{x} + (−u\underline{z})) + \underline{z}) ≫ \underline{x} = ((\underline{x} + (−u\underline{z})) + \underline{z}); eqAddition ⊳ \underline{x} = ((\underline{x} + (−u\underline{z})) + \underline{z}) ≫ (\underline{x} + \underline{y}) = (((\underline{x} + (−u\underline{z})) + \underline{z}) + \underline{y}); plusAssociativity ≫ (((\underline{x} + (−u\underline{z})) + \underline{z}) + \underline{y}) = ((\underline{x} + (−u\underline{z})) + (\underline{z} + \underline{y})); eqTransitivity ⊳ (\underline{x} + \underline{y}) = (((\underline{x} + (−u\underline{z})) + \underline{z}) + \underline{y}) ⊳ (((\underline{x} + (−u\underline{z}))+\underline{z})+\underline{y}) = ((\underline{x}+(−u\underline{z}))+(\underline{z}+\underline{y})) ≫ (\underline{x}+\underline{y}) = ((\underline{x}+(−u\underline{z}))+(\underline{z}+\underline{y}))⌉, p_{0}, c)]

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

[insertMiddleTerm(Sum) →^{tex} “insertMiddleTerm(Sum)”]

[insertMiddleTerm(Sum) →^{pyk} “lemma insertMiddleTerm(Sum)”]

insertTwoMiddleTerms(Sum)

[insertTwoMiddleTerms(Sum) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: insertMiddleTerm(Sum) ≫ (\underline{x} + \underline{y}) = ((\underline{x} + (−u\underline{z})) + (\underline{z} + \underline{y})); insertMiddleTerm(Sum) ≫ (\underline{z} + \underline{y}) = ((\underline{z} + (−u\underline{u})) + (\underline{u} + \underline{y})); EqAdditionLeft ⊳ (\underline{z} + \underline{y}) = ((\underline{z} + (−u\underline{u})) + (\underline{u} + \underline{y})) ≫ ((\underline{x}+(−u\underline{z}))+(\underline{z}+\underline{y})) = ((\underline{x}+(−u\underline{z}))+((\underline{z}+(−u\underline{u}))+(\underline{u}+\underline{y}))); plusAssociativity ≫ (((\underline{x} + (−u\underline{z})) + (\underline{z} + (−u\underline{u}))) + (\underline{u} + \underline{y})) = ((\underline{x} + (−u\underline{z})) + ((\underline{z} + (−u\underline{u})) + (\underline{u} + \underline{y}))); eqSymmetry ⊳ (((\underline{x} + (−u\underline{z})) + (\underline{z} + (−u\underline{u}))) + (\underline{u} + \underline{y})) = ((\underline{x}+(−u\underline{z}))+((\underline{z}+(−u\underline{u}))+(\underline{u}+\underline{y}))) ≫ ((\underline{x}+(−u\underline{z}))+((\underline{z}+(−u\underline{u}))+(\underline{u}+\underline{y}))) = (((\underline{x} + (−u\underline{z})) + (\underline{z} + (−u\underline{u}))) + (\underline{u} + \underline{y})); eqTransitivity4 ⊳ (\underline{x} + \underline{y}) = ((\underline{x} + (−u\underline{z})) + (\underline{z} + \underline{y})) ⊳ ((\underline{x} + (−u\underline{z})) + (\underline{z} + \underline{y})) = ((\underline{x} + (−u\underline{z})) + ((\underline{z} + (−u\underline{u})) + (\underline{u} + \underline{y}))) ⊳ ((\underline{x} + (−u\underline{z})) + ((\underline{z} + (−u\underline{u})) + (\underline{u} + \underline{y}))) = (((\underline{x} + (−u\underline{z})) + (\underline{z} + (−u\underline{u}))) + (\underline{u} + \underline{y})) ≫ (\underline{x} + \underline{y}) = (((\underline{x} + (−u\underline{z})) + (\underline{z} + (−u\underline{u}))) + (\underline{u} + \underline{y}))⌉, p_{0}, c)]

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

[insertTwoMiddleTerms(Sum) →^{tex} “insertTwoMiddleTerms(Sum)”]

[insertTwoMiddleTerms(Sum) →^{pyk} “lemma insertTwoMiddleTerms(Sum)”]

insertMiddleTerm(Difference)

[insertMiddleTerm(Difference) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: insertMiddleTerm(Sum) ≫ (\underline{x} + (−u\underline{y})) = ((\underline{x} + (−u(−u\underline{z}))) + ((−u\underline{z}) + (−u\underline{y}))); DoubleMinus ≫ (−u(−u\underline{z})) = \underline{z}; EqAdditionLeft ⊳ (−u(−u\underline{z})) = \underline{z} ≫ (\underline{x} + (−u(−u\underline{z}))) = (\underline{x} + \underline{z}); plusCommutativity ≫ ((−u\underline{z}) + (−u\underline{y})) = ((−u\underline{y}) + (−u\underline{z})); −x − y = −(x + y) ≫ ((−u\underline{y}) + (−u\underline{z})) = (−u(\underline{y} + \underline{z})); eqTransitivity ⊳ ((−u\underline{z}) + (−u\underline{y})) = ((−u\underline{y}) + (−u\underline{z})) ⊳ ((−u\underline{y}) + (−u\underline{z})) = (−u(\underline{y} + \underline{z})) ≫ ((−u\underline{z}) + (−u\underline{y})) = (−u(\underline{y} + \underline{z})); AddEquations ⊳ (\underline{x} + (−u(−u\underline{z}))) = (\underline{x} + \underline{z}) ⊳ ((−u\underline{z}) + (−u\underline{y})) = (−u(\underline{y} + \underline{z})) ≫ ((\underline{x} + (−u(−u\underline{z}))) + ((−u\underline{z}) + (−u\underline{y}))) = ((\underline{x} + \underline{z}) + (−u(\underline{y} + \underline{z}))); eqTransitivity ⊳ (\underline{x} + (−u\underline{y})) = ((\underline{x} + (−u(−u\underline{z}))) + ((−u\underline{z}) + (−u\underline{y}))) ⊳ ((\underline{x} + (−u(−u\underline{z}))) + ((−u\underline{z}) + (−u\underline{y}))) = ((\underline{x} + \underline{z}) + (−u(\underline{y} + \underline{z}))) ≫ (\underline{x} + (−u\underline{y})) = ((\underline{x} + \underline{z}) + (−u(\underline{y} + \underline{z})))⌉, p_{0}, c)]

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

[insertMiddleTerm(Difference) →^{tex} “insertMiddleTerm(Difference)”]

[insertMiddleTerm(Difference) →^{pyk} “lemma insertMiddleTerm(Difference)”]

x ∗ 0 + x = x

[x ∗ 0 + x = x →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: times1 ≫ (\underline{x} ∗ 1) = \underline{x}; eqSymmetry ⊳ (\underline{x} ∗ 1) = \underline{x} ≫ \underline{x} = (\underline{x} ∗ 1); EqAdditionLeft ⊳ \underline{x} = (\underline{x} ∗ 1) ≫ ((\underline{x} ∗ 0) + \underline{x}) = ((\underline{x} ∗ 0) + (\underline{x} ∗ 1)); Distribution ≫ (\underline{x} ∗ (0 + 1)) = ((\underline{x}∗0)+(\underline{x}∗1)); eqSymmetry⊳(\underline{x}∗(0+1)) = ((\underline{x}∗0)+(\underline{x}∗1)) ≫ ((\underline{x}∗0)+(\underline{x}∗1)) = (\underline{x} ∗ (0 + 1)); plus0Left ≫ (0 + 1) = 1; EqMultiplicationLeft ⊳ (0 + 1) = 1 ≫ (\underline{x} ∗ (0 + 1)) = (\underline{x} ∗ 1); eqTransitivity5 ⊳ ((\underline{x} ∗ 0) + \underline{x}) = ((\underline{x} ∗ 0) + (\underline{x} ∗ 1)) ⊳ ((\underline{x} ∗ 0) + (\underline{x}∗1)) = (\underline{x}∗(0+1))⊳(\underline{x}∗(0+1)) = (\underline{x}∗1)⊳(\underline{x}∗1) = \underline{x} ≫ ((\underline{x}∗0)+\underline{x}) = \underline{x}⌉, p_{0}, c)]

[x ∗ 0 + x = x →^{stmt} SystemQ ⊢ ∀\underline{x}: ((\underline{x} ∗ 0) + \underline{x}) = \underline{x}]

[x ∗ 0 + x = x →^{tex} “x∗0+x=x”]

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

x ∗ 0 = 0

[x ∗ 0 = 0 →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: x = x + (y − y) ≫ (\underline{x} ∗ 0) = ((\underline{x} ∗ 0) + (\underline{x} + (−u\underline{x}))); plusAssociativity ≫ (((\underline{x} ∗ 0) + \underline{x}) + (−u\underline{x})) = ((\underline{x} ∗ 0) + (\underline{x} + (−u\underline{x}))); eqSymmetry ⊳ (((\underline{x} ∗ 0) + \underline{x}) + (−u\underline{x})) = ((\underline{x}∗0)+(\underline{x}+(−u\underline{x}))) ≫ ((\underline{x}∗0)+(\underline{x}+(−u\underline{x}))) = (((\underline{x}∗0)+\underline{x})+(−u\underline{x})); x∗0+x = x ≫ ((\underline{x} ∗ 0) + \underline{x}) = \underline{x}; eqAddition ⊳ ((\underline{x} ∗ 0) + \underline{x}) = \underline{x} ≫ (((\underline{x} ∗ 0) + \underline{x}) + (−u\underline{x})) = (\underline{x} + (−u\underline{x})); Negative ≫ (\underline{x} + (−u\underline{x})) = 0; eqTransitivity5 ⊳ (\underline{x} ∗ 0) = ((\underline{x} ∗ 0) + (\underline{x} + (−u\underline{x}))) ⊳ ((\underline{x} ∗ 0) + (\underline{x} + (−u\underline{x}))) = (((\underline{x} ∗ 0) + \underline{x}) + (−u\underline{x})) ⊳ (((\underline{x} ∗ 0) + \underline{x}) + (−u\underline{x})) = (\underline{x} + (−u\underline{x})) ⊳ (\underline{x} + (−u\underline{x})) = 0 ≫ (\underline{x} ∗ 0) = 0⌉, p_{0}, c)] [x ∗ 0 = 0 →^{stmt} SystemQ ⊢ ∀\underline{x}: (\underline{x} ∗ 0) = 0]

[x ∗ 0 = 0 →^{tex} “x∗0=0”]

[x ∗ 0 = 0 →^{pyk} “lemma x∗0=0”]

NonnegativeFactors

[NonnegativeFactors →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ 0 <= \underline{y} ⊢ leqMultiplication⊳0 <= \underline{y}⊳0 <= \underline{x} ≫ (0∗\underline{y}) <= (\underline{x}∗\underline{y}); timesCommutativity ≫ (0∗\underline{y}) = (\underline{y}∗0); x∗0 = 0 ≫ (\underline{y}∗0) = 0; eqTransitivity⊳(0∗\underline{y}) = (\underline{y}∗0)⊳(\underline{y}∗0) = 0 ≫ (0∗\underline{y}) = 0; subLeqLeft⊳(0∗\underline{y}) = 0⊳(0∗\underline{y}) <= (\underline{x}∗\underline{y}) ≫ 0 <= (\underline{x}∗\underline{y})⌉, p_{0}, c)]

[NonnegativeFactors →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: 0 <= \underline{x} ⊢ 0 <= \underline{y} ⊢ 0 <= (\underline{x} ∗ \underline{y})]

[NonnegativeFactors →^{tex} “NonnegativeFactors”]

[NonnegativeFactors →^{pyk} “lemma nonnegativeFactors”]

NonzeroFactors

[NonzeroFactors →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ (\underline{x} = 0)n ⊢ ˙¬ (\underline{y} = 0)n ⊢ NeqMultiplication ⊳ ˙¬ (\underline{y} = 0)n ⊳ ˙¬ (\underline{x} = 0)n ≫ ˙¬ ((\underline{x} ∗ \underline{y}) = (0 ∗ \underline{y}))n; timesCommutativity ≫ (0 ∗ \underline{y}) = (\underline{y} ∗ 0); x ∗ 0 = 0 ≫ (\underline{y} ∗ 0) = 0; eqTransitivity ⊳ (0 ∗ \underline{y}) = (\underline{y} ∗ 0) ⊳ (\underline{y} ∗ 0) = 0 ≫ (0 ∗ \underline{y}) = 0; SubNeqRight ⊳ (0 ∗ \underline{y}) = 0 ⊳ ˙¬ ((\underline{x} ∗ \underline{y}) = (0 ∗ \underline{y}))n ≫ ˙¬ ((\underline{x} ∗ \underline{y}) = 0)n⌉, p_{0}, c)]

[NonzeroFactors →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ (\underline{x} = 0)n ⊢ ˙¬ (\underline{y} = 0)n ⊢ ˙¬ ((\underline{x} ∗ \underline{y}) = 0)n]

[NonzeroFactors →^{tex} “NonzeroFactors”]

[NonzeroFactors →^{pyk} “lemma nonzeroFactors”]

PositiveFactors

[PositiveFactors →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ⊢ ˙¬ (0 <= \underline{y} ⇒ ˙¬ ( ˙¬ (0 = \underline{y})n)n)n ⊢ Repetition ⊳ ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ≫ ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n; FirstConjunct ⊳ ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ≫ 0 <= \underline{x}; SecondConjunct ⊳ ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ≫ ˙¬ (0 = \underline{x})n; NeqSymmetry ⊳ ˙¬ (0 = \underline{x})n ≫ ˙¬ (\underline{x} = 0)n; Repetition ⊳ ˙¬ (0 <= \underline{y} ⇒ ˙¬ ( ˙¬ (0 = \underline{y})n)n)n ≫ ˙¬ (0 <= \underline{y} ⇒ ˙¬ ( ˙¬ (0 = \underline{y})n)n)n; FirstConjunct ⊳ ˙¬ (0 <= \underline{y} ⇒ ˙¬ ( ˙¬ (0 = \underline{y})n)n)n ≫ 0 <= \underline{y}; SecondConjunct ⊳ ˙¬ (0 <= \underline{y} ⇒ ˙¬ ( ˙¬ (0 = \underline{y})n)n)n ≫ ˙¬ (0 = \underline{y})n; NeqSymmetry ⊳ ˙¬ (0 = \underline{y})n ≫ ˙¬ (\underline{y} = 0)n; NonnegativeFactors ⊳ 0 <= \underline{x} ⊳ 0 <= \underline{y} ≫ 0 <= (\underline{x} ∗ \underline{y}); NonzeroFactors ⊳ ˙¬ (\underline{x} = 0)n ⊳ ˙¬ (\underline{y} = 0)n ≫ ˙¬ ((\underline{x} ∗ \underline{y}) = 0)n; NeqSymmetry ⊳ ˙¬ ((\underline{x} ∗ \underline{y}) = 0)n ≫ ˙¬ (0 = (\underline{x} ∗ \underline{y}))n; JoinConjuncts ⊳ 0 <= (\underline{x} ∗ \underline{y}) ⊳ ˙¬ (0 = (\underline{x} ∗ \underline{y}))n ≫ ˙¬ (0 <= (\underline{x} ∗ \underline{y}) ⇒ ˙¬ ( ˙¬ (0 = (\underline{x} ∗ \underline{y}))n)n)n; Repetition ⊳ ˙¬ (0 <= (\underline{x} ∗ \underline{y}) ⇒ ˙¬ ( ˙¬ (0 = (\underline{x} ∗ \underline{y}))n)n)n ≫ ˙¬ (0 <= (\underline{x} ∗ \underline{y}) ⇒ ˙¬ ( ˙¬ (0 = (\underline{x} ∗ \underline{y}))n)n)n⌉, p_{0}, c)]

[PositiveFactors →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ⊢ ˙¬ (0 <= \underline{y} ⇒ ˙¬ ( ˙¬ (0 = \underline{y})n)n)n ⊢ ˙¬ (0 <= (\underline{x} ∗ \underline{y}) ⇒ ˙¬ ( ˙¬ (0 = (\underline{x} ∗ \underline{y}))n)n)n]

[PositiveFactors →^{tex} “PositiveFactors”]

[PositiveFactors →^{pyk} “lemma positiveFactors”]

PlusTimesMinus

[PlusTimesMinus →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: Times(−1)Left ≫ ((−u1) ∗ \underline{y}) = (−u\underline{y}); EqMultiplicationLeft ⊳ ((−u1) ∗ \underline{y}) = (−u\underline{y}) ≫ (\underline{x} ∗ ((−u1) ∗ \underline{y})) = (\underline{x} ∗ (−u\underline{y})); timesAssociativity ≫ ((\underline{x} ∗ (−u1)) ∗ \underline{y}) = (\underline{x} ∗ ((−u1) ∗ \underline{y})); timesCommutativity ≫ (\underline{x} ∗ (−u1)) = ((−u1) ∗ \underline{x}); eqMultiplication ⊳ (\underline{x} ∗ (−u1)) = ((−u1) ∗ \underline{x}) ≫ ((\underline{x} ∗ (−u1)) ∗ \underline{y}) = (((−u1) ∗ \underline{x}) ∗ \underline{y}); timesAssociativity ≫ (((−u1) ∗ \underline{x}) ∗ \underline{y}) = ((−u1) ∗ (\underline{x} ∗ \underline{y})); Times(−1)Left ≫ ((−u1) ∗ (\underline{x} ∗ \underline{y})) = (−u(\underline{x} ∗ \underline{y})); eqTransitivity4 ⊳ ((\underline{x} ∗ (−u1)) ∗ \underline{y}) = (((−u1) ∗ \underline{x}) ∗ \underline{y}) ⊳ (((−u1) ∗ \underline{x}) ∗ \underline{y}) = ((−u1) ∗ (\underline{x} ∗ \underline{y})) ⊳ ((−u1) ∗ (\underline{x} ∗ \underline{y})) = (−u(\underline{x} ∗ \underline{y})) ≫ ((\underline{x} ∗ (−u1)) ∗ \underline{y}) = (−u(\underline{x} ∗ \underline{y})); Equality ⊳ ((\underline{x} ∗ (−u1)) ∗ \underline{y}) = (−u(\underline{x} ∗ \underline{y})) ⊳ ((\underline{x} ∗ (−u1)) ∗ \underline{y}) = (\underline{x} ∗ ((−u1) ∗ \underline{y})) ≫ (−u(\underline{x} ∗ \underline{y})) = (\underline{x}∗((−u1)∗\underline{y})); eqTransitivity⊳(−u(\underline{x}∗\underline{y})) = (\underline{x}∗((−u1)∗\underline{y}))⊳(\underline{x}∗((−u1)∗\underline{y})) = (\underline{x} ∗ (−u\underline{y})) ≫ (−u(\underline{x} ∗ \underline{y})) = (\underline{x} ∗ (−u\underline{y})); eqSymmetry ⊳ (−u(\underline{x} ∗ \underline{y})) = (\underline{x} ∗ (−u\underline{y})) ≫ (\underline{x} ∗ (−u\underline{y})) = (−u(\underline{x} ∗ \underline{y}))⌉, p_{0}, c)]

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

[PlusTimesMinus →^{tex} “PlusTimesMinus”]

[PlusTimesMinus →^{pyk} “lemma plusTimesMinus”]

MinusTimesMinus

[MinusTimesMinus →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: DoubleMinus ≫ (−u(−u\underline{y})) = \underline{y}; Times(−1)Left ≫ ((−u1) ∗ (−u\underline{y})) = (−u(−u\underline{y})); eqTransitivity ⊳ ((−u1) ∗ (−u\underline{y})) = (−u(−u\underline{y})) ⊳ (−u(−u\underline{y})) = \underline{y} ≫ ((−u1) ∗ (−u\underline{y})) = \underline{y}; EqMultiplicationLeft ⊳ ((−u1) ∗ (−u\underline{y})) = \underline{y} ≫ (\underline{x} ∗ ((−u1) ∗ (−u\underline{y}))) = (\underline{x} ∗ \underline{y}); Times(−1) ≫ (\underline{x} ∗ (−u1)) = (−u\underline{x}); eqMultiplication ⊳ (\underline{x} ∗ (−u1)) = (−u\underline{x}) ≫ ((\underline{x} ∗ (−u1)) ∗ (−u\underline{y})) = ((−u\underline{x}) ∗ (−u\underline{y})); timesAssociativity ≫ ((\underline{x} ∗ (−u1)) ∗ (−u\underline{y})) = (\underline{x} ∗ ((−u1) ∗ (−u\underline{y}))); Equality ⊳ ((\underline{x} ∗ (−u1)) ∗ (−u\underline{y})) = ((−u\underline{x})∗(−u\underline{y}))⊳((\underline{x}∗(−u1))∗(−u\underline{y})) = (\underline{x}∗((−u1)∗(−u\underline{y}))) ≫ ((−u\underline{x})∗(−u\underline{y})) = (\underline{x} ∗ ((−u1) ∗ (−u\underline{y}))); eqTransitivity ⊳ ((−u\underline{x}) ∗ (−u\underline{y})) = (\underline{x} ∗ ((−u1) ∗ (−u\underline{y}))) ⊳ (\underline{x} ∗ ((−u1) ∗ (−u\underline{y}))) = (\underline{x} ∗ \underline{y}) ≫ ((−u\underline{x}) ∗ (−u\underline{y})) = (\underline{x} ∗ \underline{y})⌉, p_{0}, c)]

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

[MinusTimesMinus →^{tex} “MinusTimesMinus”]

[MinusTimesMinus →^{pyk} “lemma minusTimesMinus”]

(−1) ∗ (−1) + (−1) ∗ 1 = 0

[(−1) ∗ (−1) + (−1) ∗ 1 = 0 →^{proof} λc.λx.P(⌈SystemQ ⊢ DistributionOut ≫ (((−u1) ∗ (−u1)) + ((−u1) ∗ 1)) = ((−u1) ∗ ((−u1) + 1)); Negative ≫ (1 + (−u1)) = 0; plusCommutativity ≫ ((−u1) + 1) = (1 + (−u1)); eqTransitivity ⊳ ((−u1) + 1) = (1 + (−u1)) ⊳ (1 + (−u1)) = 0 ≫ ((−u1) + 1) = 0; EqMultiplicationLeft ⊳ ((−u1) + 1) = 0 ≫ ((−u1)∗((−u1)+1)) = ((−u1)∗0); x∗0 = 0 ≫ ((−u1)∗0) = 0; eqTransitivity4⊳ (((−u1)∗(−u1))+((−u1)∗1)) = ((−u1)∗((−u1)+1))⊳((−u1)∗((−u1)+1)) = ((−u1) ∗ 0) ⊳ ((−u1) ∗ 0) = 0 ≫ (((−u1) ∗ (−u1)) + ((−u1) ∗ 1)) = 0⌉, p_{0}, c)]

[(−1) ∗ (−1) + (−1) ∗ 1 = 0 →^{stmt} SystemQ ⊢ (((−u1) ∗ (−u1)) + ((−u1) ∗ 1)) = 0]

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

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

(−1) ∗ (−1) = 1

[(−1) ∗ (−1) = 1 →^{proof} λc.λx.P(⌈SystemQ ⊢ x = x + (y − y) ≫ ((−u1) ∗ (−u1)) = (((−u1) ∗ (−u1)) + (1 + (−u1))); times1 ≫ ((−u1) ∗ 1) = (−u1); eqSymmetry ⊳ ((−u1) ∗ 1) = (−u1) ≫ (−u1) = ((−u1) ∗ 1); EqAdditionLeft ⊳ (−u1) = ((−u1) ∗ 1) ≫ (1 + (−u1)) = (1 + ((−u1) ∗ 1)); EqAdditionLeft ⊳ (1 + (−u1)) = (1 + ((−u1) ∗ 1)) ≫ (((−u1) ∗ (−u1)) + (1 + (−u1))) = (((−u1) ∗ (−u1)) + (1 + ((−u1) ∗ 1))); plusCommutativity ≫ (1 + ((−u1) ∗ 1)) = (((−u1) ∗ 1) + 1); EqAdditionLeft ⊳ (1 + ((−u1) ∗ 1)) = (((−u1) ∗ 1) + 1) ≫ (((−u1) ∗ (−u1)) + (1 + ((−u1) ∗ 1))) = (((−u1) ∗ (−u1)) + (((−u1) ∗ 1) + 1)); plusAssociativity ≫ ((((−u1) ∗ (−u1)) + ((−u1) ∗ 1)) + 1) = (((−u1) ∗ (−u1))+(((−u1)∗1)+1)); eqSymmetry⊳((((−u1)∗(−u1))+((−u1)∗1))+1) = (((−u1) ∗ (−u1)) + (((−u1) ∗ 1) + 1)) ≫ (((−u1) ∗ (−u1)) + (((−u1) ∗ 1) + 1)) = ((((−u1) ∗ (−u1)) + ((−u1) ∗ 1)) + 1); (−1) ∗ (−1) + (−1) ∗ 1 = 0 ≫ (((−u1)∗(−u1))+((−u1)∗1)) = 0; eqAddition⊳(((−u1)∗(−u1))+((−u1)∗1)) = 0 ≫ ((((−u1) ∗ (−u1)) + ((−u1) ∗ 1)) + 1) = (0 + 1); plus0Left ≫ (0 + 1) = 1; eqTransitivity5 ⊳ ((−u1) ∗ (−u1)) = (((−u1) ∗ (−u1)) + (1 + (−u1))) ⊳ (((−u1) ∗ (−u1)) + (1 + (−u1))) = (((−u1) ∗ (−u1)) + (1 + ((−u1) ∗ 1))) ⊳ (((−u1) ∗ (−u1)) + (1 + ((−u1) ∗ 1))) = (((−u1) ∗ (−u1)) + (((−u1) ∗ 1) + 1)) ⊳ (((−u1) ∗ (−u1)) + (((−u1) ∗ 1) + 1)) = ((((−u1) ∗ (−u1)) + ((−u1) ∗ 1)) + 1) ≫ ((−u1) ∗ (−u1)) = ((((−u1) ∗ (−u1)) + ((−u1) ∗ 1)) + 1); eqTransitivity4 ⊳ ((−u1) ∗ (−u1)) = ((((−u1) ∗ (−u1)) + ((−u1) ∗ 1)) + 1) ⊳ ((((−u1) ∗ (−u1)) + ((−u1) ∗ 1)) + 1) = (0 + 1) ⊳ (0 + 1) = 1 ≫ ((−u1) ∗ (−u1)) = 1⌉, p_{0}, c)]

[(−1) ∗ (−1) = 1 →^{stmt} SystemQ ⊢ ((−u1) ∗ (−u1)) = 1]

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

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

0 < 1Helper

[0 < 1Helper →^{proof} λc.λx.P(⌈SystemQ ⊢ 1 <= 0 ⊢ leqAddition ⊳ 1 <= 0 ≫ (1 + (−u1)) <= (0 + (−u1)); Negative ≫ (1 + (−u1)) = 0; subLeqLeft ⊳ (1 + (−u1)) = 0 ⊳ (1 + (−u1)) <= (0 + (−u1)) ≫ 0 <= (0 + (−u1)); plus0Left ≫ (0 + (−u1)) = (−u1); subLeqRight ⊳ (0 + (−u1)) = (−u1) ⊳ 0 <= (0 + (−u1)) ≫ 0 <= (−u1); leqMultiplication ⊳ 0 <= (−u1)⊳0 <= (−u1) ≫ (0∗(−u1)) <= ((−u1)∗(−u1)); x∗0 = 0 ≫ ((−u1)∗0) = 0; timesCommutativity ≫ (0∗(−u1)) = ((−u1)∗0); eqTransitivity⊳(0∗(−u1)) = ((−u1) ∗ 0) ⊳ ((−u1) ∗ 0) = 0 ≫ (0 ∗ (−u1)) = 0; subLeqLeft ⊳ (0 ∗ (−u1)) = 0 ⊳ (0 ∗ (−u1)) <= ((−u1) ∗ (−u1)) ≫ 0 <= ((−u1) ∗ (−u1)); (−1) ∗ (−1) = 1 ≫ ((−u1) ∗ (−u1)) = 1; subLeqRight ⊳ ((−u1) ∗ (−u1)) = 1 ⊳ 0 <= ((−u1)∗(−u1)) ≫ 0 <= 1; Ded⊳1 <= 0 ⊢ 0 <= 1 ≫ 1 <= 0 ⇒ 0 <= 1⌉, p_{0}, c)]

[0 < 1Helper →^{stmt} SystemQ ⊢ 1 <= 0 ⇒ 0 <= 1]

[0 < 1Helper →^{tex} “0<1Helper”]

[0 < 1Helper →^{pyk} “lemma 0<1Helper”]

0 < 1

[0 < 1 →^{proof} λc.λx.P(⌈SystemQ ⊢ leqTotality ≫ ˙¬ (0 <= 1)n ⇒ 1 <= 0; AutoImply ≫ 0 <= 1 ⇒ 0 <= 1; 0 < 1Helper ≫ 1 <= 0 ⇒ 0 <= 1; FromDisjuncts ⊳ ˙¬ (0 <= 1)n ⇒ 1 <= 0 ⊳ 0 <= 1 ⇒ 0 <= 1 ⊳ 1 <= 0 ⇒ 0 <= 1 ≫ 0 <= 1; 0not1 ≫ ˙¬ (0 = 1)n; JoinConjuncts ⊳ 0 <= 1 ⊳ ˙¬ (0 = 1)n ≫ ˙¬ (0 <= 1 ⇒ ˙¬ ( ˙¬ (0 = 1)n)n)n⌉, p_{0}, c)]

[0 < 1 →^{stmt} SystemQ ⊢ ˙¬ (0 <= 1 ⇒ ˙¬ ( ˙¬ (0 = 1)n)n)n]

[0 < 1 →^{tex} “0<1”]

[0 < 1 →^{pyk} “lemma 0<1”]

0 < 2

[0 < 2 →^{proof} λc.λx.P(⌈SystemQ ⊢ 0 < 1 ≫ ˙¬ (0 <= 1 ⇒ ˙¬ ( ˙¬ (0 = 1)n)n)n; LessAddition ⊳ ˙¬ (0 <= 1 ⇒ ˙¬ ( ˙¬ (0 = 1)n)n)n ≫ ˙¬ ((0 + 1) <= (1 + 1) ⇒ ˙¬ ( ˙¬ ((0 + 1) = (1 + 1))n)n)n; plus0Left ≫ (0 + 1) = 1; SubLessLeft ⊳ (0 + 1) = 1 ⊳ ˙¬ ((0 + 1) <= (1 + 1) ⇒ ˙¬ ( ˙¬ ((0 + 1) = (1 + 1))n)n)n ≫ ˙¬ (1 <= (1 + 1) ⇒ ˙¬ ( ˙¬ (1 = (1 + 1))n)n)n; LessTransitivity ⊳ ˙¬ (0 <= 1 ⇒ ˙¬ ( ˙¬ (0 = 1)n)n)n ⊳ ˙¬ (1 <= (1 + 1) ⇒ ˙¬ ( ˙¬ (1 = (1 + 1))n)n)n ≫ ˙¬ (0 <= (1 + 1) ⇒ ˙¬ ( ˙¬ (0 = (1 + 1))n)n)n⌉, p_{0}, c)]

[0 < 2 →^{stmt} SystemQ ⊢ ˙¬ (0 <= (1 + 1) ⇒ ˙¬ ( ˙¬ (0 = (1 + 1))n)n)n]

[0 < 2 →^{tex} “0<2”]

[0 < 2 →^{pyk} “lemma 0<2”]

0 < 3

[0 < 3 →^{proof} λc.λx.P(⌈SystemQ ⊢ 0 < 2 ≫ ˙¬ (0 <= (1 + 1) ⇒ ˙¬ ( ˙¬ (0 = (1 + 1))n)n)n; LessLeq ⊳ ˙¬ (0 <= (1 + 1) ⇒ ˙¬ ( ˙¬ (0 = (1 + 1))n)n)n ≫ 0 <= (1 + 1); Leq + 1 ⊳ 0 <= (1 + 1) ≫ ˙¬ (0 <= ((1 + 1) + 1) ⇒ ˙¬ ( ˙¬ (0 = ((1 + 1) + 1))n)n)n; Repetition ⊳ ˙¬ (0 <= ((1 + 1) + 1) ⇒ ˙¬ ( ˙¬ (0 = ((1+1)+1))n)n)n ≫ ˙¬ (0 <= ((1+1)+1) ⇒ ˙¬ ( ˙¬ (0 = ((1+1)+1))n)n)n⌉, p_{0}, c)]

[0 < 3 →^{stmt} SystemQ ⊢ ˙¬ (0 <= ((1 + 1) + 1) ⇒ ˙¬ ( ˙¬ (0 = ((1 + 1) + 1))n)n)n]

[0 < 3 →^{tex} “0<3”]

[0 < 3 →^{pyk} “lemma 0<3”]

0 < 1/2

[0 < 1/2 →^{proof} λc.λx.P(⌈SystemQ ⊢ 0 < 2 ≫ ˙¬ (0 <= (1 + 1) ⇒ ˙¬ ( ˙¬ (0 = (1 + 1))n)n)n; FirstConjunct ⊳ ˙¬ (0 <= (1 + 1) ⇒ ˙¬ ( ˙¬ (0 = (1 + 1))n)n)n ≫ 0 <= (1 + 1); SecondConjunct ⊳ ˙¬ (0 <= (1 + 1) ⇒ ˙¬ ( ˙¬ (0 = (1 + 1))n)n)n ≫ ˙¬ (0 = (1 + 1))n; NeqSymmetry ⊳ ˙¬ (0 = (1 + 1))n ≫ ˙¬ ((1 + 1) = 0)n; 0 < 1 ≫ ˙¬ (0 <= 1 ⇒ ˙¬ ( ˙¬ (0 = 1)n)n)n; x ∗ 0 = 0 ≫ ((1 + 1) ∗ 0) = 0; x ∗ y = zBackwards ⊳ ((1 + 1) ∗ 0) = 0 ≫ 0 = (0 ∗ (1 + 1)); SubLessLeft ⊳ 0 = (0 ∗ (1 + 1)) ⊳ ˙¬ (0 <= 1 ⇒ ˙¬ ( ˙¬ (0 = 1)n)n)n ≫ ˙¬ ((0 ∗ (1 + 1)) <= 1 ⇒ ˙¬ ( ˙¬ ((0 ∗ (1 + 1)) = 1)n)n)n; Reciprocal ⊳ ˙¬ ((1 + 1) = 0)n ≫ ((1 + 1) ∗ rec(1 + 1)) = 1; x ∗ y = zBackwards ⊳ ((1 + 1) ∗ rec(1 + 1)) = 1 ≫ 1 = (rec(1+1)∗(1+1)); SubLessRight⊳1 = (rec(1+1)∗(1+1))⊳ ˙¬ ((0∗(1+1)) <= 1 ⇒ ˙¬ ( ˙¬ ((0 ∗ (1 + 1)) = 1)n)n)n ≫ ˙¬ ((0 ∗ (1 + 1)) <= (rec(1 + 1) ∗ (1 + 1)) ⇒ ˙¬ ( ˙¬ ((0 ∗ (1 + 1)) = (rec(1 + 1) ∗ (1 + 1)))n)n)n; LessDivision ⊳ 0 <= (1 + 1) ⊳ ˙¬ ((0 ∗ (1 + 1)) <= (rec(1 + 1) ∗ (1 + 1)) ⇒ ˙¬ ( ˙¬ ((0 ∗ (1 + 1)) = (rec(1 + 1) ∗ (1 + 1)))n)n)n ≫ ˙¬ (0 <= rec(1 + 1) ⇒ ˙¬ ( ˙¬ (0 = rec(1 + 1))n)n)n⌉, p_{0}, c)]

[0 < 1/2 →^{stmt} SystemQ ⊢ ˙¬ (0 <= rec(1 + 1) ⇒ ˙¬ ( ˙¬ (0 = rec(1 + 1))n)n)n]

[0 < 1/2 →^{tex} “0<1/2”]

[0 < 1/2 →^{pyk} “lemma 0<1/2”]

0 < 1/3

[0 < 1/3 →^{proof} λc.λx.P(⌈SystemQ ⊢ 0 < 3 ≫ ˙¬ (0 <= ((1 + 1) + 1) ⇒ ˙¬ ( ˙¬ (0 = ((1+1)+1))n)n)n; PositiveInverted⊳ ˙¬ (0 <= ((1+1)+1) ⇒ ˙¬ ( ˙¬ (0 = ((1+1)+ 1))n)n)n ≫ ˙¬ (0 <= rec((1 + 1) + 1) ⇒ ˙¬ ( ˙¬ (0 = rec((1 + 1) + 1))n)n)n⌉, p_{0}, c)]

[0 < 1/3 →^{stmt} SystemQ ⊢ ˙¬ (0 <= rec((1 + 1) + 1) ⇒ ˙¬ ( ˙¬ (0 = rec((1 + 1) + 1))n)n)n]

[0 < 1/3 →^{tex} “0<1/3”]

[0 < 1/3 →^{pyk} “lemma 0<1/3”]

TwoWholes

[TwoWholes →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: times1 ≫ (\underline{x}∗1) = \underline{x}; eqSymmetry ≫ \underline{x} = (\underline{x}∗1); EqAdditionLeft⊳\underline{x} = (\underline{x}∗1) ≫ (\underline{x}+\underline{x}) = (\underline{x}+(\underline{x}∗1)); eqAddition⊳\underline{x} = (\underline{x} ∗ 1) ≫ (\underline{x} + (\underline{x} ∗ 1)) = ((\underline{x} ∗ 1) + (\underline{x} ∗ 1)); eqTransitivity ⊳ (\underline{x} + \underline{x}) = (\underline{x} + (\underline{x} ∗ 1)) ⊳ (\underline{x} + (\underline{x} ∗ 1)) = ((\underline{x} ∗ 1) + (\underline{x} ∗ 1)) ≫ (\underline{x} + \underline{x}) = ((\underline{x} ∗ 1) + (\underline{x} ∗ 1)); DistributionOut ≫ ((\underline{x} ∗ 1) + (\underline{x} ∗ 1)) = (\underline{x} ∗ (1 + 1)); Repetition ⊳ ((\underline{x} ∗ 1) + (\underline{x} ∗ 1)) = (\underline{x} ∗ (1 + 1)) ≫ ((\underline{x} ∗ 1) + (\underline{x} ∗ 1)) = (\underline{x} ∗ (1 + 1)); timesCommutativity ≫ (\underline{x} ∗ (1 + 1)) = ((1 + 1) ∗ \underline{x}); eqTransitivity4 ⊳ (\underline{x} + \underline{x}) = ((\underline{x} ∗ 1) + (\underline{x} ∗ 1)) ⊳ ((\underline{x} ∗ 1) + (\underline{x} ∗ 1)) = (\underline{x} ∗ (1 + 1)) ⊳ (\underline{x} ∗ (1 + 1)) = ((1 + 1) ∗ \underline{x}) ≫ (\underline{x} + \underline{x}) = ((1 + 1) ∗ \underline{x})⌉, p_{0}, c)]

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

[TwoWholes →^{tex} “TwoWholes”]

[TwoWholes →^{pyk} “lemma x+x=2∗x”]

ThreeWholes

[ThreeWholes →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: TwoWholes ≫ (\underline{x} + \underline{x}) = ((1 + 1) ∗ \underline{x}); times1Left ≫ (1 ∗ \underline{x}) = \underline{x}; eqSymmetry ⊳ (1 ∗ \underline{x}) = \underline{x} ≫ \underline{x} = (1 ∗ \underline{x}); AddEquations ⊳ (\underline{x} + \underline{x}) = ((1 + 1) ∗ \underline{x}) ⊳ \underline{x} = (1 ∗ \underline{x}) ≫ ((\underline{x} + \underline{x}) + \underline{x}) = (((1 + 1) ∗ \underline{x}) + (1 ∗ \underline{x})); DistributionOutLeft ≫ (((1 + 1) ∗ \underline{x}) + (1 ∗ \underline{x})) = (\underline{x} ∗ ((1 + 1) + 1)); timesCommutativity ≫ (\underline{x} ∗ ((1 + 1) + 1)) = (((1 + 1) + 1) ∗ \underline{x}); eqTransitivity4 ⊳ ((\underline{x} + \underline{x}) + \underline{x}) = (((1+1)∗\underline{x})+(1∗\underline{x}))⊳(((1+1)∗\underline{x})+(1∗\underline{x})) = (\underline{x}∗((1+1)+1))⊳(\underline{x}∗((1+1)+1)) = (((1 + 1) + 1) ∗ \underline{x}) ≫ ((\underline{x} + \underline{x}) + \underline{x}) = (((1 + 1) + 1) ∗ \underline{x}); Repetition ⊳ ((\underline{x} + \underline{x}) + \underline{x}) = (((1 + 1) + 1) ∗ \underline{x}) ≫ ((\underline{x} + \underline{x}) + \underline{x}) = (((1 + 1) + 1) ∗ \underline{x})⌉, p_{0}, c)]

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

[ThreeWholes →^{tex} “ThreeWholes”]

[ThreeWholes →^{pyk} “lemma x+x+x=3∗x”]

TwoHalves

[TwoHalves →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: 0 < 2 ≫ ˙¬ (0 <= (1 + 1) ⇒ ˙¬ ( ˙¬ (0 = (1 + 1))n)n)n; LessNeq ⊳ ˙¬ (0 <= (1 + 1) ⇒ ˙¬ ( ˙¬ (0 = (1 + 1))n)n)n ≫ ˙¬ (0 = (1 + 1))n; NeqSymmetry ⊳ ˙¬ (0 = (1 + 1))n ≫ ˙¬ ((1 + 1) = 0)n; TwoWholes ≫ ((rec(1+1)∗\underline{x})+(rec(1+1)∗\underline{x})) = ((1+1)∗(rec(1+1)∗\underline{x})); timesAssociativity ≫ (((1 + 1) ∗ rec(1 + 1)) ∗ \underline{x}) = ((1 + 1) ∗ (rec(1 + 1) ∗ \underline{x})); eqSymmetry ⊳ (((1 + 1) ∗ rec(1 + 1)) ∗ \underline{x}) = ((1 + 1) ∗ (rec(1 + 1) ∗ \underline{x})) ≫ ((1 + 1) ∗ (rec(1 + 1) ∗ \underline{x})) = (((1 + 1) ∗ rec(1 + 1)) ∗ \underline{x}); Reciprocal ⊳ ˙¬ ((1 + 1) = 0)n ≫ ((1 + 1) ∗ rec(1 + 1)) = 1; eqMultiplication ⊳ ((1 + 1) ∗ rec(1 + 1)) = 1 ≫ (((1 + 1) ∗ rec(1 + 1)) ∗ \underline{x}) = (1∗\underline{x}); times1Left ≫ (1∗\underline{x}) = \underline{x}; eqTransitivity5⊳((rec(1+1)∗\underline{x})+(rec(1+1)∗\underline{x})) = ((1 + 1) ∗ (rec(1 + 1) ∗ \underline{x})) ⊳ ((1 + 1) ∗ (rec(1 + 1) ∗ \underline{x})) = (((1 + 1) ∗ rec(1 + 1)) ∗ \underline{x}) ⊳ (((1 + 1) ∗ rec(1 + 1)) ∗ \underline{x}) = (1 ∗ \underline{x}) ⊳ (1 ∗ \underline{x}) = \underline{x} ≫ ((rec(1 + 1) ∗ \underline{x}) + (rec(1 + 1) ∗ \underline{x})) = \underline{x}⌉, p_{0}, c)]

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

[TwoHalves →^{tex} “TwoHalves”]

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

ThreeThirds

[ThreeThirds →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: 0 < 3 ≫ ˙¬ (0 <= ((1 + 1) + 1) ⇒ ˙¬ ( ˙¬ (0 = ((1 + 1) + 1))n)n)n; PositiveNonzero ⊳ ˙¬ (0 <= ((1 + 1) + 1) ⇒ ˙¬ ( ˙¬ (0 = ((1 + 1) + 1))n)n)n ≫ ˙¬ (((1 + 1) + 1) = 0)n; ThreeWholes ≫ (((rec((1+1)+1)∗\underline{x})+(rec((1+1)+1)∗\underline{x}))+(rec((1+1)+1)∗\underline{x})) = (((1+1)+1)∗ (rec((1+1)+1)∗\underline{x})); timesAssociativity ≫ ((((1+1)+1)∗rec((1+1)+1))∗\underline{x}) = (((1+1)+1)∗(rec((1+1)+1)∗\underline{x})); eqSymmetry⊳((((1+1)+1)∗rec((1+1)+1))∗ \underline{x}) = (((1+1)+1)∗(rec((1+1)+1)∗\underline{x})) ≫ (((1+1)+1)∗(rec((1+1)+1)∗\underline{x})) = ((((1 + 1) + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{x}); Reciprocal ⊳ ˙¬ (((1 + 1) + 1) = 0)n ≫ (((1 + 1)+1)∗rec((1+1)+1)) = 1; eqMultiplication⊳(((1+1)+1)∗rec((1+1)+1)) = 1 ≫ ((((1 + 1) + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{x}) = (1 ∗ \underline{x}); times1Left ≫ (1 ∗ \underline{x}) = \underline{x}; eqTransitivity5⊳(((rec((1+1)+1)∗\underline{x})+(rec((1+1)+1)∗\underline{x}))+(rec((1+1)+1)∗ \underline{x})) = (((1+1)+1)∗(rec((1+1)+1)∗\underline{x}))⊳(((1+1)+1)∗(rec((1+1)+1)∗\underline{x})) = ((((1 + 1) + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{x}) ⊳ ((((1 + 1) + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{x}) = (1 ∗ \underline{x}) ⊳ (1 ∗ \underline{x}) = \underline{x} ≫ (((rec((1 + 1) + 1) ∗ \underline{x}) + (rec((1 + 1) + 1) ∗ \underline{x})) + (rec((1 + 1) + 1) ∗ \underline{x})) = \underline{x}⌉, p_{0}, c)]

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

[ThreeThirds →^{tex} “ThreeThirds”]

[ThreeThirds →^{pyk} “lemma (1/3)x+(1/3)x+(1/3)x=x”]

−x − y = −(x + y)

[−x − y = −(x + y) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: Times(−1)Left ≫ ((−u1) ∗ \underline{x}) = (−u\underline{x}); Times(−1)Left ≫ ((−u1) ∗ \underline{y}) = (−u\underline{y}); AddEquations ⊳ ((−u1) ∗ \underline{x}) = (−u\underline{x}) ⊳ ((−u1) ∗ \underline{y}) = (−u\underline{y}) ≫ (((−u1) ∗ \underline{x}) + ((−u1) ∗ \underline{y})) = ((−u\underline{x})+(−u\underline{y})); eqSymmetry ⊳(((−u1)∗\underline{x})+((−u1)∗\underline{y})) = ((−u\underline{x})+(−u\underline{y})) ≫ ((−u\underline{x}) + (−u\underline{y})) = (((−u1) ∗ \underline{x}) + ((−u1) ∗ \underline{y})); DistributionOut ≫ (((−u1) ∗ \underline{x}) + ((−u1) ∗ \underline{y})) = ((−u1) ∗ (\underline{x} + \underline{y})); Times(−1)Left ≫ ((−u1) ∗ (\underline{x} + \underline{y})) = (−u(\underline{x} + \underline{y})); eqTransitivity4 ⊳ ((−u\underline{x}) + (−u\underline{y})) = (((−u1) ∗ \underline{x}) + ((−u1) ∗ \underline{y})) ⊳ (((−u1) ∗ \underline{x}) + ((−u1) ∗ \underline{y})) = ((−u1) ∗ (\underline{x} + \underline{y})) ⊳ ((−u1) ∗ (\underline{x} + \underline{y})) = (−u(\underline{x} + \underline{y})) ≫ ((−u\underline{x}) + (−u\underline{y})) = (−u(\underline{x} + \underline{y}))⌉, p_{0}, c)]

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

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

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

−x ∗ y = −(x ∗ y)

[−x ∗ y = −(x ∗ y) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: Times(−1)Left ≫ ((−u1) ∗ \underline{x}) = (−u\underline{x}); eqMultiplication ⊳ ((−u1) ∗ \underline{x}) = (−u\underline{x}) ≫ (((−u1) ∗ \underline{x}) ∗ \underline{y}) = ((−u\underline{x}) ∗ \underline{y}); eqSymmetry ⊳ (((−u1) ∗ \underline{x}) ∗ \underline{y}) = ((−u\underline{x}) ∗ \underline{y}) ≫ ((−u\underline{x}) ∗ \underline{y}) = (((−u1) ∗ \underline{x}) ∗ \underline{y}); timesAssociativity ≫ (((−u1) ∗ \underline{x}) ∗ \underline{y}) = ((−u1) ∗ (\underline{x} ∗ \underline{y})); Times(−1)Left ≫ ((−u1) ∗ (\underline{x} ∗ \underline{y})) = (−u(\underline{x} ∗ \underline{y})); eqTransitivity4 ⊳ ((−u\underline{x}) ∗ \underline{y}) = (((−u1) ∗ \underline{x}) ∗ \underline{y}) ⊳ (((−u1) ∗ \underline{x}) ∗ \underline{y}) = ((−u1)∗(\underline{x}∗\underline{y}))⊳((−u1)∗(\underline{x}∗\underline{y})) = (−u(\underline{x}∗\underline{y})) ≫ ((−u\underline{x})∗\underline{y}) = (−u(\underline{x}∗\underline{y}))⌉, p_{0}, c)] [−x ∗ y = −(x ∗ y) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ((−u\underline{x}) ∗ \underline{y}) = (−u(\underline{x} ∗ \underline{y}))] [−x ∗ y = −(x ∗ y) →^{tex} “-x∗y=-(x∗y)”] [−x ∗ y = −(x ∗ y) →^{pyk} “lemma -x∗y=-(x∗y)”]

−0 = 0

[−0 = 0 →^{proof} λc.λx.P(⌈SystemQ ⊢ Negative ≫ (0 + (−u0)) = 0; plus0 ≫ (0+0) = 0; UniqueNegative⊳(0+(−u0)) = 0⊳(0+0) = 0 ≫ (−u0) = 0⌉, p_{0}, c)] [−0 = 0 →^{stmt} SystemQ ⊢ (−u0) = 0]

[−0 = 0 →^{tex} “-0=0”]

[−0 = 0 →^{pyk} “lemma -0=0”]

SFsymmetry

[SFsymmetry →^{tex} “SFsymmetry”] [SFsymmetry →^{pyk} “lemma sameFsymmetry”]

SFtransitivity

[SFtransitivity →^{tex} “SFtransitivity”] [SFtransitivity →^{pyk} “lemma sameFtransitivity”]

f2R(Plus)

[f2R(Plus) →^{tex} “f2R(Plus)”] [f2R(Plus) →^{pyk} “lemma f2R(Plus)”]

f2R(Times)

[f2R(Times) →^{tex} “f2R(Times)”] [f2R(Times) →^{pyk} “lemma f2R(Times)”]

<< TransitivityHelper(Q)

[<< TransitivityHelper(Q) →^{tex} “<<TransitivityHelper(Q)”]

[<< TransitivityHelper(Q) →^{pyk} “lemma <<TransitivityHelper(Q)”]

<< Transitivity

[<< Transitivity →^{tex} “<<Transitivity”]

[<< Transitivity →^{pyk} “lemma <<Transitivity”]

<<== Reflexivity

[<<== Reflexivity →^{tex} “<<==Reflexivity”]

[<<== Reflexivity →^{pyk} “lemma <<==Reflexivity”]

<<== AntisymmetryHelper(Q)

[<<== AntisymmetryHelper(Q) →^{tex} “<<==AntisymmetryHelper(Q)”]

[<<== AntisymmetryHelper(Q) →^{pyk} “lemma <<==AntisymmetryHelper(Q)”]

FromNot < f(Weak)(Helper)

[FromNot < f(Weak)(Helper) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ ( ˙¬ ( ˙¬ (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 ⊢ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⊢ FromNegatedAnd ⊳ ˙¬ ( ˙¬ ( ˙¬ (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 ⊳ ˙¬ (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; FromNegatedImply ⊳ ˙¬ (\underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})))n ≫ ˙¬ (\underline{n} <= \underline{m} ⇒ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})))n)n)n; FirstConjunct ⊳ ˙¬ (\underline{n} <= \underline{m} ⇒ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})))n)n)n ≫ \underline{n} <= \underline{m}; SecondConjunct ⊳ ˙¬ (\underline{n} <= \underline{m} ⇒ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})))n)n)n ≫ ˙¬ (\underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})))n; ToLess ⊳ ˙¬ (\underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})))n ≫ ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})) <= \underline{(fx)}[\underline{m}] ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})) = \underline{(fx)}[\underline{m}])n)n)n; JoinConjuncts ⊳ \underline{n} <= \underline{m} ⊳ ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})) <= \underline{(fx)}[\underline{m}] ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})) = \underline{(fx)}[\underline{m}])n)n)n ≫ ˙¬ (\underline{n} <= \underline{m} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})) <= \underline{(fx)}[\underline{m}] ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})) = \underline{(fx)}[\underline{m}])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)}: ˙¬ ( ˙¬ ( ˙¬ (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 ⊢ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⊢ ˙¬ (\underline{n} <= \underline{m} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})) <= \underline{(fx)}[\underline{m}] ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})) = \underline{(fx)}[\underline{m}])n)n)n)n)n ≫ ˙¬ ( ˙¬ ( ˙¬ (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 ⇒ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})) <= \underline{(fx)}[\underline{m}] ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})) = \underline{(fx)}[\underline{m}])n)n)n)n)n⌉, p_{0}, c)]

[FromNot < f(Weak)(Helper) →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ ( ˙¬ ( ˙¬ (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 ⇒ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})) <= \underline{(fx)}[\underline{m}] ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})) = \underline{(fx)}[\underline{m}])n)n)n)n)n]

[FromNot < f(Weak)(Helper) →^{tex} “FromNot<f(Weak)(Helper)”]

[FromNot < f(Weak)(Helper) →^{pyk} “lemma fromNot<f(Weak) helper”]

FromNot < f(Weak)

[FromNot < f(Weak) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ (∀_{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 ⊢ 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}] <= (\underline{(fy)}[\underline{m}] + (−u\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}] <= (\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; ∀\underline{m}: ∀\underline{n}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: Ded ⊳ ∀\underline{m}: ∀\underline{n}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ (∀_{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 ⊢ ˙¬ (∀_{obj}\overline{(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 ⇒ ˙¬ (∀_{obj}\overline{(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 ⊢ MT ⊳ ˙¬ (∀_{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 ⇒ ˙¬ (∀_{obj}\overline{(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}\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)n; EEA − negated ⊳ ˙¬ ( ˙¬ (∀_{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)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; A4 @ \underline{(ϵ)} ⊳ ∀_{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 ≫ ∀_{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; A4 @ \underline{n} ⊳ ∀_{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 ≫ ˙¬ (∀_{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; FromNot < f(Weak)(Helper) ≫ ˙¬ ( ˙¬ ( ˙¬ (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 ⇒ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})) <= \underline{(fx)}[\underline{m}] ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})) = \underline{(fx)}[\underline{m}])n)n)n)n)n; ExistMP ⊳ ˙¬ ( ˙¬ ( ˙¬ (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 ⇒ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})) <= \underline{(fx)}[\underline{m}] ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})) = \underline{(fx)}[\underline{m}])n)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)n)n)n ≫ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})) <= \underline{(fx)}[\underline{m}] ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})) = \underline{(fx)}[\underline{m}])n)n)n)n)n; IntroExist @ \underline{m} ⊳ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})) <= \underline{(fx)}[\underline{m}] ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})) = \underline{(fx)}[\underline{m}])n)n)n)n)n ≫ ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})) <= \underline{(fx)}[\underline{m}] ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})) = \underline{(fx)}[\underline{m}])n)n)n)n)n)n)n; Gen ⊳ ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})) <= \underline{(fx)}[\underline{m}] ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})) = \underline{(fx)}[\underline{m}])n)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{(fy)}[\underline{m}] + (−u\underline{(ϵ)})) <= \underline{(fx)}[\underline{m}] ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})) = \underline{(fx)}[\underline{m}])n)n)n)n)n)n)n; Gen ⊳ ∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})) <= \underline{(fx)}[\underline{m}] ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})) = \underline{(fx)}[\underline{m}])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{(fy)}[\underline{m}] + (−u\underline{(ϵ)})) <= \underline{(fx)}[\underline{m}] ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})) = \underline{(fx)}[\underline{m}])n)n)n)n)n)n)n⌉, p_{0}, c)]

[FromNot < f(Weak) →^{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)n ⊢ ∀_{obj}\underline{(ϵ)}: ∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})) <= \underline{(fx)}[\underline{m}] ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})) = \underline{(fx)}[\underline{m}])n)n)n)n)n)n)n]

[FromNot < f(Weak) →^{tex} “FromNot<f(Weak)”]

[FromNot < f(Weak) →^{pyk} “lemma fromNot<f(Weak)”]

FromNot < f(Strong)(Helper2)

[FromNot < f(Strong)(Helper2) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: ∀\underline{v}: ˙¬ (|(\underline{x}+(−u\underline{y}))| <= (rec((1+1)+1)∗\underline{v}) ⇒ ˙¬ ( ˙¬ (|(\underline{x}+(−u\underline{y}))| = (rec((1 + 1) + 1) ∗ \underline{v}))n)n)n ⊢ ˙¬ (|(\underline{z} + (−u\underline{u}))| <= (rec((1 + 1) + 1) ∗ \underline{v}) ⇒ ˙¬ ( ˙¬ (|(\underline{z} + (−u\underline{u}))| = (rec((1 + 1) + 1) ∗ \underline{v}))n)n)n ⊢ ˙¬ ((\underline{y} + (−u\underline{u})) <= (rec((1 + 1) + 1) ∗ \underline{v}) ⇒ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{u})) = (rec((1 + 1) + 1) ∗ \underline{v}))n)n)n ⊢ x <= |x| ≫ (\underline{x} + (−u\underline{y})) <= |(\underline{x} + (−u\underline{y}))|; leqLessTransitivity ⊳ (\underline{x} + (−u\underline{y})) <= |(\underline{x} + (−u\underline{y}))| ⊳ ˙¬ (|(\underline{x} + (−u\underline{y}))| <= (rec((1 + 1) + 1) ∗ \underline{v}) ⇒ ˙¬ ( ˙¬ (|(\underline{x} + (−u\underline{y}))| = (rec((1 + 1) + 1) ∗ \underline{v}))n)n)n ≫ ˙¬ ((\underline{x} + (−u\underline{y})) <= (rec((1 + 1) + 1) ∗ \underline{v}) ⇒ ˙¬ ( ˙¬ ((\underline{x} + (−u\underline{y})) = (rec((1 + 1) + 1) ∗ \underline{v}))n)n)n; NumericalDifference ≫ |(\underline{z} + (−u\underline{u}))| = |(\underline{u} + (−u\underline{z}))|; SubLessLeft ⊳ |(\underline{z} + (−u\underline{u}))| = |(\underline{u} + (−u\underline{z}))| ⊳ ˙¬ (|(\underline{z} + (−u\underline{u}))| <= (rec((1 + 1) + 1) ∗ \underline{v}) ⇒ ˙¬ ( ˙¬ (|(\underline{z} + (−u\underline{u}))| = (rec((1 + 1) + 1) ∗ \underline{v}))n)n)n ≫ ˙¬ (|(\underline{u} + (−u\underline{z}))| <= (rec((1 + 1) + 1) ∗ \underline{v}) ⇒ ˙¬ ( ˙¬ (|(\underline{u}+(−u\underline{z}))| = (rec((1+1)+1)∗\underline{v}))n)n)n; x <= |x| ≫ (\underline{u}+(−u\underline{z})) <= |(\underline{u}+ (−u\underline{z}))|; leqLessTransitivity⊳(\underline{u}+(−u\underline{z})) <= |(\underline{u}+(−u\underline{z}))|⊳ ˙¬ (|(\underline{u}+(−u\underline{z}))| <= (rec((1 + 1) + 1) ∗ \underline{v}) ⇒ ˙¬ ( ˙¬ (|(\underline{u} + (−u\underline{z}))| = (rec((1 + 1) + 1) ∗ \underline{v}))n)n)n ≫ ˙¬ ((\underline{u} + (−u\underline{z})) <= (rec((1 + 1) + 1) ∗ \underline{v}) ⇒ ˙¬ ( ˙¬ ((\underline{u} + (−u\underline{z})) = (rec((1 + 1) + 1) ∗ \underline{v}))n)n)n; AddEquations(Less) ⊳ ˙¬ ((\underline{x} + (−u\underline{y})) <= (rec((1 + 1) + 1) ∗ \underline{v}) ⇒ ˙¬ ( ˙¬ ((\underline{x} + (−u\underline{y})) = (rec((1 + 1) + 1) ∗ \underline{v}))n)n)n ⊳ ˙¬ ((\underline{y} + (−u\underline{u})) <= (rec((1 + 1) + 1) ∗ \underline{v}) ⇒ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{u})) = (rec((1 + 1) + 1) ∗ \underline{v}))n)n)n ≫ ˙¬ (((\underline{x}+(−u\underline{y}))+(\underline{y}+(−u\underline{u}))) <= ((rec((1+1)+1)∗\underline{v})+(rec((1+1)+1)∗\underline{v})) ⇒ ˙¬ ( ˙¬ (((\underline{x} + (−u\underline{y})) + (\underline{y} + (−u\underline{u}))) = ((rec((1 + 1) + 1) ∗ \underline{v}) + (rec((1 + 1) + 1) ∗ \underline{v})))n)n)n; AddEquations(Less) ⊳ ˙¬ (((\underline{x} + (−u\underline{y})) + (\underline{y} + (−u\underline{u}))) <= ((rec((1+1)+1)∗\underline{v})+(rec((1+1)+1)∗\underline{v})) ⇒ ˙¬ ( ˙¬ (((\underline{x}+(−u\underline{y}))+(\underline{y}+(−u\underline{u}))) = ((rec((1 + 1) + 1) ∗ \underline{v}) + (rec((1 + 1) + 1) ∗ \underline{v})))n)n)n ⊳ ˙¬ ((\underline{u} + (−u\underline{z})) <= (rec((1 + 1) + 1) ∗ \underline{v}) ⇒ ˙¬ ( ˙¬ ((\underline{u} + (−u\underline{z})) = (rec((1 + 1) + 1) ∗ \underline{v}))n)n)n ≫ ˙¬ ((((\underline{x} + (−u\underline{y})) + (\underline{y} + (−u\underline{u}))) + (\underline{u} + (−u\underline{z}))) <= (((rec((1+1)+1)∗\underline{v})+(rec((1+1)+1)∗\underline{v}))+(rec((1+1)+1)∗\underline{v})) ⇒ ˙¬ ( ˙¬ ((((\underline{x}+ (−u\underline{y}))+(\underline{y} +(−u\underline{u})))+(\underline{u}+(−u\underline{z}))) = (((rec((1+1)+1)∗\underline{v})+(rec((1+1)+1)∗ \underline{v}))+(rec((1+1)+1)∗\underline{v})))n)n)n; insertTwoMiddleTerms(Sum) ≫ (\underline{x}+(−u\underline{z})) = (((\underline{x} + (−u\underline{y})) + (\underline{y} + (−u\underline{u}))) + (\underline{u} + (−u\underline{z}))); eqSymmetry ⊳ (\underline{x} + (−u\underline{z})) = (((\underline{x}+(−u\underline{y}))+(\underline{y} +(−u\underline{u})))+(\underline{u}+(−u\underline{z}))) ≫ (((\underline{x}+(−u\underline{y}))+(\underline{y} +(−u\underline{u})))+(\underline{u}+ (−u\underline{z}))) = (\underline{x}+(−u\underline{z})); SubLessLeft⊳(((\underline{x}+(−u\underline{y}))+(\underline{y}+(−u\underline{u})))+(\underline{u}+(−u\underline{z}))) = (\underline{x} + (−u\underline{z})) ⊳ ˙¬ ((((\underline{x} + (−u\underline{y})) + (\underline{y} + (−u\underline{u}))) + (\underline{u} + (−u\underline{z}))) <= (((rec((1 + 1) + 1) ∗ \underline{v}) + (rec((1 + 1) + 1) ∗ \underline{v})) + (rec((1 + 1) + 1) ∗ \underline{v})) ⇒ ˙¬ ( ˙¬ ((((\underline{x} + (−u\underline{y})) + (\underline{y} + (−u\underline{u}))) + (\underline{u} + (−u\underline{z}))) = (((rec((1 + 1) + 1) ∗ \underline{v}) + (rec((1 + 1) + 1) ∗ \underline{v})) + (rec((1 + 1) + 1) ∗ \underline{v})))n)n)n ≫ ˙¬ ((\underline{x} + (−u\underline{z})) <= (((rec((1 + 1) + 1) ∗ \underline{v}) + (rec((1 + 1) + 1) ∗ \underline{v})) + (rec((1 + 1) + 1) ∗ \underline{v})) ⇒ ˙¬ ( ˙¬ ((\underline{x} + (−u\underline{z})) = (((rec((1 + 1) + 1) ∗ \underline{v}) + (rec((1 + 1) + 1) ∗ \underline{v})) + (rec((1 + 1) + 1) ∗ \underline{v})))n)n)n; ThreeThirds ≫ (((rec((1 + 1) + 1) ∗ \underline{v}) + (rec((1 + 1) + 1) ∗ \underline{v})) + (rec((1 + 1) + 1) ∗ \underline{v})) = \underline{v}; SubLessRight ⊳ (((rec((1 + 1) + 1) ∗ \underline{v}) + (rec((1 + 1) + 1) ∗ \underline{v})) + (rec((1 + 1) + 1) ∗ \underline{v})) = \underline{v} ⊳ ˙¬ ((\underline{x} + (−u\underline{z})) <= (((rec((1 + 1) + 1) ∗ \underline{v}) + (rec((1 + 1) + 1) ∗ \underline{v})) + (rec((1 + 1) + 1) ∗ \underline{v})) ⇒ ˙¬ ( ˙¬ ((\underline{x} + (−u\underline{z})) = (((rec((1 + 1) + 1) ∗ \underline{v}) + (rec((1 + 1) + 1) ∗ \underline{v})) + (rec((1 + 1) + 1) ∗ \underline{v})))n)n)n ≫ ˙¬ ((\underline{x} + (−u\underline{z})) <= \underline{v} ⇒ ˙¬ ( ˙¬ ((\underline{x} + (−u\underline{z})) = \underline{v})n)n)n⌉, p_{0}, c)]

[FromNot < f(Strong)(Helper2) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: ∀\underline{v}: ˙¬ (|(\underline{x} + (−u\underline{y}))| <= (rec((1 + 1) + 1) ∗ \underline{v}) ⇒ ˙¬ ( ˙¬ (|(\underline{x} + (−u\underline{y}))| = (rec((1 + 1) + 1) ∗ \underline{v}))n)n)n ⊢ ˙¬ (|(\underline{z} + (−u\underline{u}))| <= (rec((1 + 1) + 1) ∗ \underline{v}) ⇒ ˙¬ ( ˙¬ (|(\underline{z} + (−u\underline{u}))| = (rec((1 + 1) + 1) ∗ \underline{v}))n)n)n ⊢ ˙¬ ((\underline{y} + (−u\underline{u})) <= (rec((1 + 1) + 1) ∗ \underline{v}) ⇒ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{u})) = (rec((1+1)+1)∗\underline{v}))n)n)n ⊢ ˙¬ ((\underline{x}+(−u\underline{z})) <= \underline{v} ⇒ ˙¬ ( ˙¬ ((\underline{x}+(−u\underline{z})) = \underline{v})n)n)n]

[FromNot < f(Strong)(Helper2) →^{tex} “FromNot<f(Strong)(Helper2)”]

[FromNot < f(Strong)(Helper2) →^{pyk} “lemma fromNot<f(Strong) helper2”]

FromNot < f(Strong)(Helper)

[FromNot < f(Strong)(Helper) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{m}: ∀\underline{(n1)}: ∀\underline{(n2)}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ (0 <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{(n2)}]+(−u(rec((1+1)+1)∗\underline{(ϵ)}))) <= \underline{(fx)}[\underline{(n2)}] ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{(n2)}]+ (−u(rec((1 + 1) + 1) ∗ \underline{(ϵ)}))) = \underline{(fx)}[\underline{(n2)}])n)n)n)n)n ⊢ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1+1)+1)∗\underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}]+(−u\underline{(fx)}[\underline{(v2)}]))| <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n)n)n ⊢ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⊢ \underline{(n2)} <= \underline{m} ⊢ 0 < 3 ≫ ˙¬ (0 <= ((1 + 1) + 1) ⇒ ˙¬ ( ˙¬ (0 = ((1 + 1) + 1))n)n)n; PositiveInverted ⊳ ˙¬ (0 <= ((1 + 1) + 1) ⇒ ˙¬ ( ˙¬ (0 = ((1 + 1) + 1))n)n)n ≫ ˙¬ (0 <= rec((1 + 1) + 1) ⇒ ˙¬ ( ˙¬ (0 = rec((1 + 1) + 1))n)n)n; PositiveFactors ⊳ ˙¬ (0 <= rec((1 + 1) + 1) ⇒ ˙¬ ( ˙¬ (0 = rec((1+1)+1))n)n)n⊳ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ≫ ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n; MP ⊳ ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{(n2)}] + (−u(rec((1 + 1) + 1) ∗ \underline{(ϵ)}))) <= \underline{(fx)}[\underline{(n2)}] ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{(n2)}] + (−u(rec((1 + 1) + 1) ∗ \underline{(ϵ)}))) = \underline{(fx)}[\underline{(n2)}])n)n)n)n)n ⊳ ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ≫ ˙¬ (\underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{(n2)}] + (−u(rec((1 + 1) + 1) ∗ \underline{(ϵ)}))) <= \underline{(fx)}[\underline{(n2)}] ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{(n2)}] + (−u(rec((1 + 1) + 1) ∗ \underline{(ϵ)}))) = \underline{(fx)}[\underline{(n2)}])n)n)n)n)n; FirstConjunct ⊳ ˙¬ (\underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{(n2)}] + (−u(rec((1 + 1) + 1) ∗ \underline{(ϵ)}))) <= \underline{(fx)}[\underline{(n2)}] ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{(n2)}]+(−u(rec((1+1)+1)∗\underline{(ϵ)}))) = \underline{(fx)}[\underline{(n2)}])n)n)n)n)n ≫ \underline{(n1)} <= \underline{(n2)}; SecondConjunct ⊳ ˙¬ (\underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{(n2)}] + (−u(rec((1 + 1) + 1) ∗ \underline{(ϵ)}))) <= \underline{(fx)}[\underline{(n2)}] ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{(n2)}] + (−u(rec((1 + 1) + 1) ∗ \underline{(ϵ)}))) = \underline{(fx)}[\underline{(n2)}])n)n)n)n)n ≫ ˙¬ ((\underline{(fy)}[\underline{(n2)}] + (−u(rec((1 + 1) + 1) ∗ \underline{(ϵ)}))) <= \underline{(fx)}[\underline{(n2)}] ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{(n2)}] + (−u(rec((1 + 1) + 1) ∗ \underline{(ϵ)}))) = \underline{(fx)}[\underline{(n2)}])n)n)n; SwitchTerms(x − y < z) ⊳ ˙¬ ((\underline{(fy)}[\underline{(n2)}] + (−u(rec((1 + 1) + 1) ∗ \underline{(ϵ)}))) <= \underline{(fx)}[\underline{(n2)}] ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{(n2)}] + (−u(rec((1 + 1) + 1) ∗ \underline{(ϵ)}))) = \underline{(fx)}[\underline{(n2)}])n)n)n ≫ ˙¬ ((\underline{(fy)}[\underline{(n2)}] + (−u\underline{(fx)}[\underline{(n2)}])) <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{(n2)}]+(−u\underline{(fx)}[\underline{(n2)}])) = (rec((1+1)+1)∗\underline{(ϵ)}))n)n)n; leqTransitivity⊳ \underline{(n1)} <= \underline{(n2)} ⊳ \underline{(n2)} <= \underline{m} ≫ \underline{(n1)} <= \underline{m}; A4 @ \underline{m} ⊳ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1+1)+1)∗\underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}]+(−u\underline{(fx)}[\underline{(v2)}]))| <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n)n)n ≫ ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1+1)+1)∗\underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{m} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fx)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{m}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{m}] + (−u\underline{(fy)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n)n)n; A4 @ \underline{(n2)} ⊳ ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1+1)+1)∗\underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{m} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fx)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{m}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{m}] + (−u\underline{(fy)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n)n)n ≫ ˙¬ (0 <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1+1)+1)∗\underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{m} ⇒ \underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fx)}[\underline{(n2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fx)}[\underline{(n2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{m}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{m}] + (−u\underline{(fy)}[\underline{(n2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n)n)n; MP3 ⊳ ˙¬ (0 <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1+1)+1)∗\underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{m} ⇒ \underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fx)}[\underline{(n2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fx)}[\underline{(n2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{m}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{m}] + (−u\underline{(fy)}[\underline{(n2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n)n)n ⊳ ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⊳ \underline{(n1)} <= \underline{m} ⊳ \underline{(n1)} <= \underline{(n2)} ≫ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fx)}[\underline{(n2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fx)}[\underline{(n2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{m}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{m}] + (−u\underline{(fy)}[\underline{(n2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n)n)n; FirstConjunct ⊳ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fx)}[\underline{(n2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fx)}[\underline{(n2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{m}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{m}] + (−u\underline{(fy)}[\underline{(n2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n)n)n ≫ ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fx)}[\underline{(n2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fx)}[\underline{(n2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n; SecondConjunct ⊳ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fx)}[\underline{(n2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fx)}[\underline{(n2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{m}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{m}] + (−u\underline{(fy)}[\underline{(n2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n)n)n ≫ ˙¬ (|(\underline{(fy)}[\underline{m}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{m}] + (−u\underline{(fy)}[\underline{(n2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n; FromNot < f(Strong)(Helper2) ⊳ ˙¬ (|(\underline{(fy)}[\underline{m}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{m}] + (−u\underline{(fy)}[\underline{(n2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⊳ ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fx)}[\underline{(n2)}]))| <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}]+(−u\underline{(fx)}[\underline{(n2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⊳ ˙¬ ((\underline{(fy)}[\underline{(n2)}] + (−u\underline{(fx)}[\underline{(n2)}])) <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{(n2)}] + (−u\underline{(fx)}[\underline{(n2)}])) = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ≫ ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) = \underline{(ϵ)})n)n)n; ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{m}: ∀\underline{(n1)}: ∀\underline{(n2)}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: Ded ⊳ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{m}: ∀\underline{(n1)}: ∀\underline{(n2)}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ (0 <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{(n2)}]+(−u(rec((1+1)+1)∗\underline{(ϵ)}))) <= \underline{(fx)}[\underline{(n2)}] ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{(n2)}]+ (−u(rec((1 + 1) + 1) ∗ \underline{(ϵ)}))) = \underline{(fx)}[\underline{(n2)}])n)n)n)n)n ⊢ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1+1)+1)∗\underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}]+(−u\underline{(fx)}[\underline{(v2)}]))| <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}]+(−u\underline{(fy)}[\underline{(v2)}]))| = (rec((1+1)+1)∗\underline{(ϵ)}))n)n)n)n)n ⊢ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⊢ \underline{(n2)} <= \underline{m} ⊢ ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) = \underline{(ϵ)})n)n)n ≫ ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{(n2)}]+(−u(rec((1+1)+1)∗\underline{(ϵ)}))) <= \underline{(fx)}[\underline{(n2)}] ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{(n2)}]+ (−u(rec((1+1)+1)∗\underline{(ϵ)}))) = \underline{(fx)}[\underline{(n2)}])n)n)n)n)n ⇒ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1+1)+1)∗\underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}]+(−u\underline{(fx)}[\underline{(v2)}]))| <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}]+(−u\underline{(fy)}[\underline{(v2)}]))| = (rec((1+1)+1)∗\underline{(ϵ)}))n)n)n)n)n ⇒ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n2)} <= \underline{m} ⇒ ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) = \underline{(ϵ)})n)n)n⌉, p_{0}, c)]

[FromNot < f(Strong)(Helper) →^{stmt} SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{m}: ∀\underline{(n1)}: ∀\underline{(n2)}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ (0 <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{(n2)}]+(−u(rec((1+1)+1)∗\underline{(ϵ)}))) <= \underline{(fx)}[\underline{(n2)}] ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{(n2)}]+ (−u(rec((1+1)+1)∗\underline{(ϵ)}))) = \underline{(fx)}[\underline{(n2)}])n)n)n)n)n ⇒ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1+1)+1)∗\underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}]+(−u\underline{(fx)}[\underline{(v2)}]))| <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n)n)n ⇒ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n2)} <= \underline{m} ⇒ ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) = \underline{(ϵ)})n)n)n]

[FromNot < f(Strong)(Helper) →^{tex} “FromNot<f(Strong)(Helper)”]

[FromNot < f(Strong)(Helper) →^{pyk} “lemma fromNot<f(Strong) helper”]

FromNot < f(Strong)

[FromNot < f(Strong) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{m}: ∀\underline{(n2)}: ∀\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)n ⊢ FromNot < f(Weak) ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(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}\underline{(ϵ)}: ∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{(n2)}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{(n2)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{(n2)}] + (−u\underline{(ϵ)})) <= \underline{(fx)}[\underline{(n2)}] ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{(n2)}] + (−u\underline{(ϵ)})) = \underline{(fx)}[\underline{(n2)}])n)n)n)n)n)n)n; A4 @(rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⊳ ∀_{obj}\underline{(ϵ)}: ∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{(n2)}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{(n2)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{(n2)}] + (−u\underline{(ϵ)})) <= \underline{(fx)}[\underline{(n2)}] ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{(n2)}] + (−u\underline{(ϵ)})) = \underline{(fx)}[\underline{(n2)}])n)n)n)n)n)n)n ≫ ∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{(n2)}: ˙¬ ( ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ (\underline{n} <= \underline{(n2)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{(n2)}] + (−u(rec((1 + 1) + 1) ∗ \underline{(ϵ)}))) <= \underline{(fx)}[\underline{(n2)}] ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{(n2)}] + (−u(rec((1 + 1) + 1) ∗ \underline{(ϵ)}))) = \underline{(fx)}[\underline{(n2)}])n)n)n)n)n)n)n; A4 @ \underline{(n1)} ⊳ ∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{(n2)}: ˙¬ ( ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ (\underline{n} <= \underline{(n2)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{(n2)}] + (−u(rec((1 + 1) + 1) ∗ \underline{(ϵ)}))) <= \underline{(fx)}[\underline{(n2)}] ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{(n2)}] + (−u(rec((1 + 1) + 1) ∗ \underline{(ϵ)}))) = \underline{(fx)}[\underline{(n2)}])n)n)n)n)n)n)n ≫ ˙¬ (∀_{obj}\underline{(n2)}: ˙¬ ( ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{(n2)}]+(−u(rec((1+1)+1)∗\underline{(ϵ)}))) <= \underline{(fx)}[\underline{(n2)}] ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{(n2)}]+ (−u(rec((1 + 1) + 1) ∗ \underline{(ϵ)}))) = \underline{(fx)}[\underline{(n2)}])n)n)n)n)n)n)n; Cauchy(2) ≫ ∀_{obj}\underline{(ϵ)}: ˙¬ (∀_{obj}\underline{(n1)}: ˙¬ (∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n)n)n)n)n; A4 @(rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⊳ ∀_{obj}\underline{(ϵ)}: ˙¬ (∀_{obj}\underline{(n1)}: ˙¬ (∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n)n)n)n)n ≫ ˙¬ (∀_{obj}\underline{(n1)}: ˙¬ (∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n)n)n)n)n; FromNot < f(Strong)(Helper) ≫ ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{(n2)}] + (−u(rec((1 + 1) + 1) ∗ \underline{(ϵ)}))) <= \underline{(fx)}[\underline{(n2)}] ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{(n2)}] + (−u(rec((1 + 1) + 1) ∗ \underline{(ϵ)}))) = \underline{(fx)}[\underline{(n2)}])n)n)n)n)n ⇒ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}]+(−u\underline{(fy)}[\underline{(v2)}]))| = (rec((1+1)+1)∗\underline{(ϵ)}))n)n)n)n)n ⇒ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n2)} <= \underline{m} ⇒ ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) = \underline{(ϵ)})n)n)n; ExistMP2 ⊳ ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{(n2)}] + (−u(rec((1 + 1) + 1) ∗ \underline{(ϵ)}))) <= \underline{(fx)}[\underline{(n2)}] ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{(n2)}] + (−u(rec((1 + 1) + 1) ∗ \underline{(ϵ)}))) = \underline{(fx)}[\underline{(n2)}])n)n)n)n)n ⇒ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}]+(−u\underline{(fy)}[\underline{(v2)}]))| = (rec((1+1)+1)∗\underline{(ϵ)}))n)n)n)n)n ⇒ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n2)} <= \underline{m} ⇒ ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) = \underline{(ϵ)})n)n)n ⊳ ˙¬ (∀_{obj}\underline{(n2)}: ˙¬ ( ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{(n2)}] + (−u(rec((1 + 1) + 1) ∗ \underline{(ϵ)}))) <= \underline{(fx)}[\underline{(n2)}] ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{(n2)}] + (−u(rec((1 + 1) + 1) ∗ \underline{(ϵ)}))) = \underline{(fx)}[\underline{(n2)}])n)n)n)n)n)n)n ⊳ ˙¬ (∀_{obj}\underline{(n1)}: ˙¬ (∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1+1)+1)∗\underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}]+(−u\underline{(fx)}[\underline{(v2)}]))| <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n)n)n)n)n ≫ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n2)} <= \underline{m} ⇒ ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) = \underline{(ϵ)})n)n)n⌉, p_{0}, c)]

[FromNot < f(Strong) →^{stmt} SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{m}: ∀\underline{(n2)}: ∀\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)n ⊢ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n2)} <= \underline{m} ⇒ ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) = \underline{(ϵ)})n)n)n]

[FromNot < f(Strong) →^{tex} “FromNot<f(Strong)”]

[FromNot < f(Strong) →^{pyk} “lemma fromNot<f(Strong)”]

fromNotSameF(Strongest)(Helper2)

[fromNotSameF(Strongest)(Helper2) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: ∀\underline{v}: ˙¬ (|(\underline{x} + (−u\underline{y}))| <= (rec((1 + 1) + 1) ∗ \underline{v}) ⇒ ˙¬ ( ˙¬ (|(\underline{x} + (−u\underline{y}))| = (rec((1 + 1) + 1) ∗ \underline{v}))n)n)n ⊢ ˙¬ (|(\underline{z} + (−u\underline{u}))| <= (rec((1 + 1) + 1) ∗ \underline{v}) ⇒ ˙¬ ( ˙¬ (|(\underline{z} + (−u\underline{u}))| = (rec((1 + 1) + 1) ∗ \underline{v}))n)n)n ⊢ \underline{v} <= |(\underline{y} + (−u\underline{u}))| ⊢ NumericalDifference ≫ |(\underline{x} + (−u\underline{y}))| = |(\underline{y} + (−u\underline{x}))|; SubLessLeft ⊳ |(\underline{x} + (−u\underline{y}))| = |(\underline{y} + (−u\underline{x}))| ⊳ ˙¬ (|(\underline{x} + (−u\underline{y}))| <= (rec((1 + 1) + 1) ∗ \underline{v}) ⇒ ˙¬ ( ˙¬ (|(\underline{x} + (−u\underline{y}))| = (rec((1 + 1) + 1) ∗ \underline{v}))n)n)n ≫ ˙¬ (|(\underline{y} + (−u\underline{x}))| <= (rec((1 + 1) + 1) ∗ \underline{v}) ⇒ ˙¬ ( ˙¬ (|(\underline{y} + (−u\underline{x}))| = (rec((1 + 1) +

1) ∗ \underline{v}))n)n)n; LessNegated ⊳ ˙¬ (|(\underline{y} + (−u\underline{x}))| <= (rec((1 + 1) + 1) ∗ \underline{v}) ⇒ ˙¬ ( ˙¬ (|(\underline{y}+(−u\underline{x}))| = (rec((1+1)+1)∗\underline{v}))n)n)n ≫ ˙¬ ((−u(rec((1+1)+1)∗\underline{v})) <= (−u|(\underline{y} + (−u\underline{x}))|) ⇒ ˙¬ ( ˙¬ ((−u(rec((1 + 1) + 1) ∗ \underline{v})) = (−u|(\underline{y}+(−u\underline{x}))|))n)n)n; LessNegated⊳ ˙¬ (|(\underline{z}+(−u\underline{u}))| <= (rec((1+1)+1)∗\underline{v}) ⇒ ˙¬ ( ˙¬ (|(\underline{z}+(−u\underline{u}))| = (rec((1+1)+1)∗\underline{v}))n)n)n ≫ ˙¬ ((−u(rec((1+1)+1)∗\underline{v})) <= (−u|(\underline{z} + (−u\underline{u}))|) ⇒ ˙¬ ( ˙¬ ((−u(rec((1 + 1) + 1) ∗ \underline{v})) = (−u|(\underline{z} + (−u\underline{u}))|))n)n)n; AddEquations(LeqLess) ⊳ \underline{v} <= |(\underline{y} + (−u\underline{u}))| ⊳ ˙¬ ((−u(rec((1 + 1) + 1) ∗ \underline{v})) <= (−u|(\underline{y} + (−u\underline{x}))|) ⇒ ˙¬ ( ˙¬ ((−u(rec((1 + 1) + 1) ∗ \underline{v})) = (−u|(\underline{y} + (−u\underline{x}))|))n)n)n ≫ ˙¬ ((\underline{v} + (−u(rec((1 + 1) + 1) ∗ \underline{v}))) <= (|(\underline{y} + (−u\underline{u}))| + (−u|(\underline{y} + (−u\underline{x}))|)) ⇒ ˙¬ ( ˙¬ ((\underline{v} + (−u(rec((1 + 1) + 1) ∗ \underline{v}))) = (|(\underline{y} + (−u\underline{u}))| + (−u|(\underline{y} + (−u\underline{x}))|)))n)n)n; AddEquations(Less) ⊳ ˙¬ ((\underline{v} + (−u(rec((1 + 1) + 1) ∗ \underline{v}))) <= (|(\underline{y} + (−u\underline{u}))| + (−u|(\underline{y} + (−u\underline{x}))|)) ⇒ ˙¬ ( ˙¬ ((\underline{v} + (−u(rec((1 + 1) + 1) ∗ \underline{v}))) = (|(\underline{y} + (−u\underline{u}))| + (−u|(\underline{y} + (−u\underline{x}))|)))n)n)n ⊳ ˙¬ ((−u(rec((1 + 1) + 1) ∗ \underline{v})) <= (−u|(\underline{z}+(−u\underline{u}))|) ⇒ ˙¬ ( ˙¬ ((−u(rec((1+1)+1)∗\underline{v})) = (−u|(\underline{z}+(−u\underline{u}))|))n)n)n ≫ ˙¬ (((\underline{v} + (−u(rec((1 + 1) + 1) ∗ \underline{v}))) + (−u(rec((1 + 1) + 1) ∗ \underline{v}))) <= ((|(\underline{y} + (−u\underline{u}))| + (−u|(\underline{y} + (−u\underline{x}))|)) + (−u|(\underline{z} + (−u\underline{u}))|)) ⇒ ˙¬ ( ˙¬ (((\underline{v} + (−u(rec((1 + 1) + 1) ∗ \underline{v}))) + (−u(rec((1 + 1) + 1) ∗ \underline{v}))) = ((|(\underline{y} + (−u\underline{u}))| + (−u|(\underline{y} + (−u\underline{x}))|)) + (−u|(\underline{z} + (−u\underline{u}))|)))n)n)n; insertTwoMiddleTerms(Numerical) ≫ |(\underline{y} + (−u\underline{u}))| <= ((|(\underline{y} + (−u\underline{x}))| + |(\underline{x} + (−u\underline{z}))|) + |(\underline{z} + (−u\underline{u}))|); plusAssociativity ≫ ((|(\underline{y} + (−u\underline{x}))| + |(\underline{x} + (−u\underline{z}))|) + |(\underline{z} + (−u\underline{u}))|) = (|(\underline{y} + (−u\underline{x}))| + (|(\underline{x} + (−u\underline{z}))| + |(\underline{z} + (−u\underline{u}))|)); plusCommutativity ≫ (|(\underline{y}+(−u\underline{x}))|+(|(\underline{x}+(−u\underline{z}))|+|(\underline{z}+(−u\underline{u}))|)) = ((|(\underline{x}+(−u\underline{z}))|+|(\underline{z}+(−u\underline{u}))|)+ |(\underline{y} + (−u\underline{x}))|); eqTransitivity ⊳ ((|(\underline{y} + (−u\underline{x}))| + |(\underline{x} + (−u\underline{z}))|) + |(\underline{z} + (−u\underline{u}))|) = (|(\underline{y} + (−u\underline{x}))| + (|(\underline{x} + (−u\underline{z}))| + |(\underline{z} + (−u\underline{u}))|)) ⊳ (|(\underline{y} + (−u\underline{x}))| + (|(\underline{x} + (−u\underline{z}))| + |(\underline{z} + (−u\underline{u}))|)) = ((|(\underline{x} + (−u\underline{z}))| + |(\underline{z} + (−u\underline{u}))|) + |(\underline{y} + (−u\underline{x}))|) ≫ ((|(\underline{y}+(−u\underline{x}))|+|(\underline{x}+(−u\underline{z}))|)+|(\underline{z}+(−u\underline{u}))|) = ((|(\underline{x}+(−u\underline{z}))|+|(\underline{z}+(−u\underline{u}))|)+ |(\underline{y} + (−u\underline{x}))|); subLeqRight ⊳ ((|(\underline{y} + (−u\underline{x}))| + |(\underline{x} + (−u\underline{z}))|) + |(\underline{z} + (−u\underline{u}))|) = ((|(\underline{x} + (−u\underline{z}))| + |(\underline{z} + (−u\underline{u}))|) + |(\underline{y} + (−u\underline{x}))|) ⊳ |(\underline{y} + (−u\underline{u}))| <= ((|(\underline{y} + (−u\underline{x}))| + |(\underline{x} + (−u\underline{z}))|) + |(\underline{z} + (−u\underline{u}))|) ≫ |(\underline{y} + (−u\underline{u}))| <= ((|(\underline{x} + (−u\underline{z}))| + |(\underline{z} + (−u\underline{u}))|) + |(\underline{y} + (−u\underline{x}))|); PositiveToLeft(Leq) ⊳ |(\underline{y} + (−u\underline{u}))| <= ((|(\underline{x} + (−u\underline{z}))| + |(\underline{z} + (−u\underline{u}))|) + |(\underline{y} + (−u\underline{x}))|) ≫ (|(\underline{y} + (−u\underline{u}))| + (−u|(\underline{y} + (−u\underline{x}))|)) <= (|(\underline{x} + (−u\underline{z}))| + |(\underline{z} + (−u\underline{u}))|); PositiveToLeft(Leq) ⊳ (|(\underline{y} + (−u\underline{u}))| + (−u|(\underline{y} + (−u\underline{x}))|)) <= (|(\underline{x} + (−u\underline{z}))| + |(\underline{z} + (−u\underline{u}))|) ≫ ((|(\underline{y} + (−u\underline{u}))| + (−u|(\underline{y} + (−u\underline{x}))|)) + (−u|(\underline{z} + (−u\underline{u}))|)) <= |(\underline{x} + (−u\underline{z}))|; LessLeqTransitivity ⊳ ˙¬ (((\underline{v} + (−u(rec((1 + 1) + 1) ∗ \underline{v}))) + (−u(rec((1 + 1) + 1) ∗ \underline{v}))) <= ((|(\underline{y} + (−u\underline{u}))| + (−u|(\underline{y} + (−u\underline{x}))|)) + (−u|(\underline{z} + (−u\underline{u}))|)) ⇒ ˙¬ ( ˙¬ (((\underline{v} + (−u(rec((1 + 1) + 1) ∗ \underline{v}))) + (−u(rec((1 + 1) + 1) ∗ \underline{v}))) = ((|(\underline{y} + (−u\underline{u}))| + (−u|(\underline{y} + (−u\underline{x}))|)) + (−u|(\underline{z} + (−u\underline{u}))|)))n)n)n ⊳ ((|(\underline{y} + (−u\underline{u}))| + (−u|(\underline{y} + (−u\underline{x}))|)) + (−u|(\underline{z} + (−u\underline{u}))|)) <= |(\underline{x} + (−u\underline{z}))| ≫ ˙¬ (((\underline{v}+(−u(rec((1+1)+1)∗\underline{v})))+(−u(rec((1+1)+1)∗\underline{v}))) <= |(\underline{x}+(−u\underline{z}))| ⇒ ˙¬ ( ˙¬ (((\underline{v} + (−u(rec((1 + 1) + 1) ∗ \underline{v}))) + (−u(rec((1 + 1) + 1) ∗ \underline{v}))) = |(\underline{x} + (−u\underline{z}))|)n)n)n; ThreeThirds ≫ (((rec((1 + 1) + 1) ∗ \underline{v}) + (rec((1 + 1) + 1) ∗ \underline{v})) + (rec((1 + 1) + 1) ∗ \underline{v})) = \underline{v}; PositiveToRight(Eq)⊳(((rec((1+1)+1)∗\underline{v})+(rec((1+1)+1)∗\underline{v}))+(rec((1+ 1)+1)∗\underline{v})) = \underline{v} ≫ ((rec((1+1)+1)∗\underline{v})+(rec((1+1)+1)∗\underline{v})) = (\underline{v}+(−u(rec((1+ 1)+1)∗\underline{v}))); PositiveToRight(Eq)⊳((rec((1+1)+1)∗\underline{v})+(rec((1+1)+1)∗\underline{v})) = (\underline{v} + (−u(rec((1 + 1) + 1) ∗ \underline{v}))) ≫ (rec((1 + 1) + 1) ∗ \underline{v}) = ((\underline{v} + (−u(rec((1 + 1) + 1) ∗ \underline{v}))) + (−u(rec((1 + 1) + 1) ∗ \underline{v}))); eqSymmetry ⊳ (rec((1 + 1) + 1) ∗ \underline{v}) = ((\underline{v} + (−u(rec((1 + 1) + 1) ∗ \underline{v}))) + (−u(rec((1 + 1) + 1) ∗ \underline{v}))) ≫ ((\underline{v} + (−u(rec((1 + 1) + 1) ∗ \underline{v}))) + (−u(rec((1 + 1) + 1) ∗ \underline{v}))) = (rec((1 + 1) + 1) ∗ \underline{v}); SubLessLeft ⊳ ((\underline{v} + (−u(rec((1 + 1) + 1) ∗ \underline{v}))) + (−u(rec((1 + 1) + 1) ∗ \underline{v}))) = (rec((1 + 1) + 1) ∗ \underline{v}) ⊳ ˙¬ (((\underline{v} + (−u(rec((1 + 1) + 1) ∗ \underline{v}))) + (−u(rec((1 + 1) + 1) ∗ \underline{v}))) <= |(\underline{x} + (−u\underline{z}))| ⇒ ˙¬ ( ˙¬ (((\underline{v} + (−u(rec((1 + 1) + 1) ∗ \underline{v}))) + (−u(rec((1 + 1) + 1) ∗ \underline{v}))) = |(\underline{x} + (−u\underline{z}))|)n)n)n ≫ ˙¬ ((rec((1 + 1) + 1) ∗ \underline{v}) <= |(\underline{x} + (−u\underline{z}))| ⇒ ˙¬ ( ˙¬ ((rec((1 + 1) + 1) ∗ \underline{v}) = |(\underline{x} + (−u\underline{z}))|)n)n)n⌉, p_{0}, c)]

[fromNotSameF(Strongest)(Helper2) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: ∀\underline{v}: ˙¬ (|(\underline{x} + (−u\underline{y}))| <= (rec((1 + 1) + 1) ∗ \underline{v}) ⇒ ˙¬ ( ˙¬ (|(\underline{x} + (−u\underline{y}))| = (rec((1 + 1) + 1) ∗ \underline{v}))n)n)n ⊢ ˙¬ (|(\underline{z} + (−u\underline{u}))| <= (rec((1 + 1) + 1) ∗ \underline{v}) ⇒ ˙¬ ( ˙¬ (|(\underline{z} + (−u\underline{u}))| = (rec((1 + 1) + 1) ∗ \underline{v}))n)n)n ⊢ \underline{v} <= |(\underline{y} + (−u\underline{u}))| ⊢ ˙¬ ((rec((1 + 1) + 1) ∗ \underline{v}) <= |(\underline{x} + (−u\underline{z}))| ⇒ ˙¬ ( ˙¬ ((rec((1 + 1) + 1) ∗ \underline{v}) = |(\underline{x} + (−u\underline{z}))|)n)n)n]

[fromNotSameF(Strongest)(Helper2) →^{tex} “fromNotSameF(Strongest)(Helper2)”]

[fromNotSameF(Strongest)(Helper2) →^{pyk} “lemma fromNotSameF(Strongest) helper2”]

fromNotSameF(Strongest)(Helper)

[fromNotSameF(Strongest)(Helper) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{m}: ∀\underline{(n1)}: ∀\underline{(n2)}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| = \underline{(ϵ)})n)n)n)n ⊢ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1+1)+1)∗\underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}]+(−u\underline{(fx)}[\underline{(v2)}]))| <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n)n)n ⊢ \underline{(n2)} <= \underline{m} ⊢ FromNegated(2 ∗ Imply) ⊳ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| = \underline{(ϵ)})n)n)n)n ≫ ˙¬ ( ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{(n2)})n)n ⇒ ˙¬ ( ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| = \underline{(ϵ)})n)n)n)n)n)n; FirstConjunct ⊳ ˙¬ ( ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{(n2)})n)n ⇒ ˙¬ ( ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| = \underline{(ϵ)})n)n)n)n)n)n ≫ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{(n2)})n)n; FirstConjunct ⊳ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{(n2)})n)n ≫ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n; SecondConjunct ⊳ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{(n2)})n)n ≫ \underline{(n1)} <= \underline{(n2)}; SecondConjunct ⊳ ˙¬ ( ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{(n2)})n)n ⇒ ˙¬ ( ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| = \underline{(ϵ)})n)n)n)n)n)n ≫ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| = \underline{(ϵ)})n)n)n)n; fromNotLess ⊳ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| = \underline{(ϵ)})n)n)n)n ≫ \underline{(ϵ)} <= |(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))|; A4 @ \underline{m} ⊳ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1+1)+1)∗\underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}]+(−u\underline{(fx)}[\underline{(v2)}]))| <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n)n)n ≫ ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1+1)+1)∗\underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{m} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fx)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{m}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{m}] + (−u\underline{(fy)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n)n)n; A4 @ \underline{(n2)} ⊳ ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1+1)+1)∗\underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{m} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fx)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{m}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{m}] + (−u\underline{(fy)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n)n)n ≫ ˙¬ (0 <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1+1)+1)∗\underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{m} ⇒ \underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fx)}[\underline{(n2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fx)}[\underline{(n2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{m}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{m}] + (−u\underline{(fy)}[\underline{(n2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n)n)n; 0 < 1/3 ≫ ˙¬ (0 <= rec((1 + 1) + 1) ⇒ ˙¬ ( ˙¬ (0 = rec((1 + 1) + 1))n)n)n; PositiveFactors ⊳ ˙¬ (0 <= rec((1 + 1) + 1) ⇒ ˙¬ ( ˙¬ (0 = rec((1 + 1) + 1))n)n)n ⊳ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ≫ ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n; leqTransitivity ⊳ \underline{(n1)} <= \underline{(n2)} ⊳ \underline{(n2)} <= \underline{m} ≫ \underline{(n1)} <= \underline{m}; MP3 ⊳ ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{m} ⇒ \underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fx)}[\underline{(n2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fx)}[\underline{(n2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{m}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{m}] + (−u\underline{(fy)}[\underline{(n2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n)n)n ⊳ ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⊳ \underline{(n1)} <= \underline{m} ⊳ \underline{(n1)} <= \underline{(n2)} ≫ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fx)}[\underline{(n2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fx)}[\underline{(n2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{m}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{m}] + (−u\underline{(fy)}[\underline{(n2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n)n)n; FirstConjunct ⊳ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fx)}[\underline{(n2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fx)}[\underline{(n2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{m}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{m}] + (−u\underline{(fy)}[\underline{(n2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n)n)n ≫ ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fx)}[\underline{(n2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fx)}[\underline{(n2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n; SecondConjunct ⊳ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fx)}[\underline{(n2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fx)}[\underline{(n2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{m}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{m}] + (−u\underline{(fy)}[\underline{(n2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n)n)n ≫ ˙¬ (|(\underline{(fy)}[\underline{m}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{m}] + (−u\underline{(fy)}[\underline{(n2)}]))| = (rec((1+1)+1)∗\underline{(ϵ)}))n)n)n; fromNotSameF(Strongest)(Helper2)⊳ ˙¬ (|(\underline{(fx)}[\underline{m}]+ (−u\underline{(fx)}[\underline{(n2)}]))| <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}]+(−u\underline{(fx)}[\underline{(n2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⊳ ˙¬ (|(\underline{(fy)}[\underline{m}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{m}] + (−u\underline{(fy)}[\underline{(n2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n⊳\underline{(ϵ)} <= |(\underline{(fx)}[\underline{(n2)}]+(−u\underline{(fy)}[\underline{(n2)}]))| ≫ ˙¬ ((rec((1+1)+1)∗\underline{(ϵ)}) <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ⇒ ˙¬ ( ˙¬ ((rec((1 + 1) + 1) ∗ \underline{(ϵ)}) = |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|)n)n)n; ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{(n1)}: ∀\underline{(n2)}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(n2)}]+(−u\underline{(fy)}[\underline{(n2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}]+(−u\underline{(fy)}[\underline{(n2)}]))| = \underline{(ϵ)})n)n)n)n ⊢ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n)n)n ⊢ FromNegatedImply ⊳ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(n2)}]+(−u\underline{(fy)}[\underline{(n2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}]+(−u\underline{(fy)}[\underline{(n2)}]))| = \underline{(ϵ)})n)n)n)n ≫ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ ( ˙¬ (\underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(n2)}]+(−u\underline{(fy)}[\underline{(n2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}]+(−u\underline{(fy)}[\underline{(n2)}]))| = \underline{(ϵ)})n)n)n)n)n)n; FirstConjunct ⊳ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ ( ˙¬ (\underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| = \underline{(ϵ)})n)n)n)n)n)n ≫ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n; 0 < 1/3 ≫ ˙¬ (0 <= rec((1 + 1) + 1) ⇒ ˙¬ ( ˙¬ (0 = rec((1 + 1) + 1))n)n)n; PositiveFactors ⊳ ˙¬ (0 <= rec((1 + 1) + 1) ⇒ ˙¬ ( ˙¬ (0 = rec((1 + 1) + 1))n)n)n ⊳ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ≫ ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n; ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{m}: ∀\underline{(n1)}: ∀\underline{(n2)}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: Ded ⊳ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{m}: ∀\underline{(n1)}: ∀\underline{(n2)}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| = \underline{(ϵ)})n)n)n)n ⊢ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1+1)+1)∗\underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}]+(−u\underline{(fx)}[\underline{(v2)}]))| <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}]+(−u\underline{(fy)}[\underline{(v2)}]))| = (rec((1+1)+1)∗\underline{(ϵ)}))n)n)n)n)n ⊢ \underline{(n2)} <= \underline{m} ⊢ ˙¬ ((rec((1 + 1) + 1) ∗ \underline{(ϵ)}) <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ⇒ ˙¬ ( ˙¬ ((rec((1 + 1) + 1) ∗ \underline{(ϵ)}) = |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|)n)n)n ≫ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| = \underline{(ϵ)})n)n)n)n ⇒ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1+1)+1)∗\underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}]+(−u\underline{(fx)}[\underline{(v2)}]))| <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n)n)n ⇒ \underline{(n2)} <= \underline{m} ⇒ ˙¬ ((rec((1 + 1) + 1) ∗ \underline{(ϵ)}) <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ⇒ ˙¬ ( ˙¬ ((rec((1 + 1) + 1) ∗ \underline{(ϵ)}) = |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|)n)n)n; Ded ⊳ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{(n1)}: ∀\underline{(n2)}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| = \underline{(ϵ)})n)n)n)n ⊢ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n)n)n ⊢ ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ≫ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| = \underline{(ϵ)})n)n)n)n ⇒ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1+1)+1)∗\underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}]+(−u\underline{(fx)}[\underline{(v2)}]))| <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n)n)n ⇒ ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n; JoinConjuncts(2conditions) ⊳ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| = \underline{(ϵ)})n)n)n)n ⇒ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n)n)n ⇒ ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⊳ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| = \underline{(ϵ)})n)n)n)n ⇒ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1+1)+1)∗\underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}]+(−u\underline{(fx)}[\underline{(v2)}]))| <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}]+(−u\underline{(fy)}[\underline{(v2)}]))| = (rec((1+1)+1)∗\underline{(ϵ)}))n)n)n)n)n ⇒ \underline{(n2)} <= \underline{m} ⇒ ˙¬ ((rec((1 + 1) + 1) ∗ \underline{(ϵ)}) <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ⇒ ˙¬ ( ˙¬ ((rec((1 + 1) + 1) ∗ \underline{(ϵ)}) = |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|)n)n)n ≫ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| = \underline{(ϵ)})n)n)n)n ⇒ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1+1)+1)∗\underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}]+(−u\underline{(fx)}[\underline{(v2)}]))| <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n)n)n ⇒ ˙¬ ( ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ (\underline{(n2)} <= \underline{m} ⇒ ˙¬ ((rec((1 + 1) + 1) ∗ \underline{(ϵ)}) <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ⇒ ˙¬ ( ˙¬ ((rec((1 + 1) + 1) ∗ \underline{(ϵ)}) = |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|)n)n)n)n)n⌉, p_{0}, c)]

[fromNotSameF(Strongest)(Helper) →^{stmt} SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{m}: ∀\underline{(n1)}: ∀\underline{(n2)}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| = \underline{(ϵ)})n)n)n)n ⇒ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1+1)+1)∗\underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}]+(−u\underline{(fx)}[\underline{(v2)}]))| <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n)n)n ⇒ ˙¬ ( ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ (\underline{(n2)} <= \underline{m} ⇒ ˙¬ ((rec((1 + 1) + 1) ∗ \underline{(ϵ)}) <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ⇒ ˙¬ ( ˙¬ ((rec((1 + 1) + 1) ∗ \underline{(ϵ)}) = |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|)n)n)n)n)n]

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

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

fromNotSameF(Strongest)

[fromNotSameF(Strongest) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{m}: ∀\underline{n}: ∀\underline{(n1)}: ∀\underline{(n2)}: ∀\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 ⊢ 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; 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}] + (−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}\underline{(ϵ)}: ˙¬ (∀_{obj}\underline{(n1)}: ˙¬ (∀_{obj}\underline{(n2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(n2)}]+(−u\underline{(fy)}[\underline{(n2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}]+(−u\underline{(fy)}[\underline{(n2)}]))| = \underline{(ϵ)})n)n)n)n)n)n; AEA − negated ⊳ ˙¬ (∀_{obj}\underline{(ϵ)}: ˙¬ (∀_{obj}\underline{(n1)}: ˙¬ (∀_{obj}\underline{(n2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(n2)}]+(−u\underline{(fy)}[\underline{(n2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}]+(−u\underline{(fy)}[\underline{(n2)}]))| = \underline{(ϵ)})n)n)n)n)n)n ≫ ˙¬ (∀_{obj}\underline{(ϵ)}: ˙¬ (∀_{obj}\underline{(n1)}: ˙¬ (∀_{obj}\underline{(n2)}: ˙¬ ( ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| = \underline{(ϵ)})n)n)n)n)n)n)n)n; Cauchy(2) ≫ ∀_{obj}\underline{(ϵ)}: ˙¬ (∀_{obj}\underline{(n1)}: ˙¬ (∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n)n)n)n)n; A4 @(rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⊳ ∀_{obj}\underline{(ϵ)}: ˙¬ (∀_{obj}\underline{(n1)}: ˙¬ (∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n)n)n)n)n ≫ ˙¬ (∀_{obj}\underline{(n1)}: ˙¬ (∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n)n)n)n)n; fromNotSameF(Strongest)(Helper) ≫ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(n2)}]+(−u\underline{(fy)}[\underline{(n2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}]+(−u\underline{(fy)}[\underline{(n2)}]))| = \underline{(ϵ)})n)n)n)n ⇒ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n)n)n ⇒ ˙¬ ( ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ (\underline{(n2)} <= \underline{m} ⇒ ˙¬ ((rec((1 + 1) + 1) ∗ \underline{(ϵ)}) <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ⇒ ˙¬ ( ˙¬ ((rec((1 + 1) + 1) ∗ \underline{(ϵ)}) = |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|)n)n)n)n)n; EAE − MP ⊳ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| = \underline{(ϵ)})n)n)n)n ⇒ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1+1)+1)∗\underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}]+(−u\underline{(fx)}[\underline{(v2)}]))| <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n)n)n ⇒ ˙¬ ( ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ (\underline{(n2)} <= \underline{m} ⇒ ˙¬ ((rec((1 + 1) + 1) ∗ \underline{(ϵ)}) <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ⇒ ˙¬ ( ˙¬ ((rec((1 + 1) + 1) ∗ \underline{(ϵ)}) = |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|)n)n)n)n)n ⊳ ˙¬ (∀_{obj}\underline{(ϵ)}: ˙¬ (∀_{obj}\underline{(n1)}: ˙¬ (∀_{obj}\underline{(n2)}: ˙¬ ( ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| = \underline{(ϵ)})n)n)n)n)n)n)n)n ≫ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = (rec((1+1)+1)∗\underline{(ϵ)}))n)n)n)n)n ⇒ ˙¬ ( ˙¬ (0 <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ (\underline{(n2)} <= \underline{m} ⇒ ˙¬ ((rec((1 + 1) + 1) ∗ \underline{(ϵ)}) <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ⇒ ˙¬ ( ˙¬ ((rec((1 + 1) + 1) ∗ \underline{(ϵ)}) = |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|)n)n)n)n)n; ExistMP ⊳ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = (rec((1+1)+1)∗\underline{(ϵ)}))n)n)n)n)n ⇒ ˙¬ ( ˙¬ (0 <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ (\underline{(n2)} <= \underline{m} ⇒ ˙¬ ((rec((1 + 1) + 1) ∗ \underline{(ϵ)}) <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ⇒ ˙¬ ( ˙¬ ((rec((1 + 1) + 1) ∗ \underline{(ϵ)}) = |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|)n)n)n)n)n ⊳ ˙¬ (∀_{obj}\underline{(n1)}: ˙¬ (∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1+1)+1)∗\underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}]+(−u\underline{(fx)}[\underline{(v2)}]))| <= (rec((1+1)+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n)n)n)n)n ≫ ˙¬ ( ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ (\underline{(n2)} <= \underline{m} ⇒ ˙¬ ((rec((1 + 1) + 1) ∗ \underline{(ϵ)}) <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ⇒ ˙¬ ( ˙¬ ((rec((1 + 1) + 1) ∗ \underline{(ϵ)}) = |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|)n)n)n)n)n; Gen ⊳ ˙¬ ( ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ (\underline{(n2)} <= \underline{m} ⇒ ˙¬ ((rec((1 + 1) + 1) ∗ \underline{(ϵ)}) <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ⇒ ˙¬ ( ˙¬ ((rec((1 + 1) + 1) ∗ \underline{(ϵ)}) = |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|)n)n)n)n)n ≫ ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ (\underline{(n2)} <= \underline{m} ⇒ ˙¬ ((rec((1 + 1) + 1) ∗ \underline{(ϵ)}) <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ⇒ ˙¬ ( ˙¬ ((rec((1 + 1) + 1) ∗ \underline{(ϵ)}) = |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|)n)n)n)n)n; IntroExist @ \underline{(n2)} ⊳ ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ (\underline{(n2)} <= \underline{m} ⇒ ˙¬ ((rec((1 + 1) + 1) ∗ \underline{(ϵ)}) <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ⇒ ˙¬ ( ˙¬ ((rec((1 + 1) + 1) ∗ \underline{(ϵ)}) = |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|)n)n)n)n)n ≫ ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{m} ⇒ ˙¬ ((rec((1 + 1) + 1) ∗ \underline{(ϵ)}) <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ⇒ ˙¬ ( ˙¬ ((rec((1 + 1) + 1) ∗ \underline{(ϵ)}) = |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|)n)n)n)n)n)n)n; IntroExist @(rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⊳ ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{m} ⇒ ˙¬ ((rec((1 + 1) + 1) ∗ \underline{(ϵ)}) <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ⇒ ˙¬ ( ˙¬ ((rec((1 + 1) + 1) ∗ \underline{(ϵ)}) = |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|)n)n)n)n)n)n)n ≫ ˙¬ (∀_{obj}\underline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\underline{(n1)}: ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{m} ⇒ ˙¬ (\underline{(ϵ)} <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ⇒ ˙¬ ( ˙¬ (\underline{(ϵ)} = |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|)n)n)n)n)n)n)n)n)n⌉, p_{0}, c)]

[fromNotSameF(Strongest) →^{stmt} SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{m}: ∀\underline{n}: ∀\underline{(n1)}: ∀\underline{(n2)}: ∀\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 ⊢ ˙¬ (∀_{obj}\underline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\underline{(n1)}: ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{m} ⇒ ˙¬ (\underline{(ϵ)} <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ⇒ ˙¬ ( ˙¬ (\underline{(ϵ)} = |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|)n)n)n)n)n)n)n)n)n] [fromNotSameF(Strongest) →^{tex} “fromNotSameF(Strongest)”]

[fromNotSameF(Strongest) →^{pyk} “lemma fromNotSameF(Strongest)”]

ToLess(F)(Helper)

[ToLess(F)(Helper) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{(n1)}: ∀\underline{(n2)}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{m} ⇒ ˙¬ (\underline{(ϵ)} <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ⇒ ˙¬ ( ˙¬ (\underline{(ϵ)} = |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|)n)n)n)n)n ⊢ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n2)} <= \underline{m} ⇒ ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) = \underline{(ϵ)})n)n)n ⊢ if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{m} ⊢ MaxLeq(1) ≫ \underline{(n1)} <= if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}); leqTransitivity ⊳ \underline{(n1)} <= if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) ⊳ if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{m} ≫ \underline{(n1)} <= \underline{m}; MaxLeq(2) ≫ \underline{(n2)} <= if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}); leqTransitivity ⊳ \underline{(n2)} <= if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) ⊳ if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{m} ≫ \underline{(n2)} <= \underline{m}; A4 @ \underline{m} ⊳ ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{m} ⇒ ˙¬ (\underline{(ϵ)} <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ⇒ ˙¬ ( ˙¬ (\underline{(ϵ)} = |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|)n)n)n)n)n ≫ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{m} ⇒ ˙¬ (\underline{(ϵ)} <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ⇒ ˙¬ ( ˙¬ (\underline{(ϵ)} = |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|)n)n)n)n)n; FirstConjunct ⊳ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{m} ⇒ ˙¬ (\underline{(ϵ)} <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ⇒ ˙¬ ( ˙¬ (\underline{(ϵ)} = |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|)n)n)n)n)n ≫ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n; SecondConjunct ⊳ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{m} ⇒ ˙¬ (\underline{(ϵ)} <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ⇒ ˙¬ ( ˙¬ (\underline{(ϵ)} = |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|)n)n)n)n)n ≫ \underline{(n1)} <= \underline{m} ⇒ ˙¬ (\underline{(ϵ)} <= |(\underline{(fx)}[\underline{m}]+(−u\underline{(fy)}[\underline{m}]))| ⇒ ˙¬ ( ˙¬ (\underline{(ϵ)} = |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|)n)n)n; MP ⊳ \underline{(n1)} <= \underline{m} ⇒ ˙¬ (\underline{(ϵ)} <= |(\underline{(fx)}[\underline{m}]+(−u\underline{(fy)}[\underline{m}]))| ⇒ ˙¬ ( ˙¬ (\underline{(ϵ)} = |(\underline{(fx)}[\underline{m}]+(−u\underline{(fy)}[\underline{m}]))|)n)n)n⊳\underline{(n1)} <= \underline{m} ≫ ˙¬ (\underline{(ϵ)} <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ⇒ ˙¬ ( ˙¬ (\underline{(ϵ)} = |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|)n)n)n; FromNumericalGreater ⊳ ˙¬ (\underline{(ϵ)} <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ⇒ ˙¬ ( ˙¬ (\underline{(ϵ)} = |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|)n)n)n ≫ ˙¬ ( ˙¬ ((\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}])) <= (−u\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ ((\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}])) = (−u\underline{(ϵ)}))n)n)n)n ⇒ ˙¬ (\underline{(ϵ)} <= (\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}])) ⇒ ˙¬ ( ˙¬ (\underline{(ϵ)} = (\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}])))n)n)n; MP2 ⊳ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n2)} <= \underline{m} ⇒ ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) = \underline{(ϵ)})n)n)n ⊳ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⊳ \underline{(n2)} <= \underline{m} ≫ ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) = \underline{(ϵ)})n)n)n; LessNegated ⊳ ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) = \underline{(ϵ)})n)n)n ≫ ˙¬ ((−u\underline{(ϵ)}) <= (−u(\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}]))) ⇒ ˙¬ ( ˙¬ ((−u\underline{(ϵ)}) = (−u(\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}]))))n)n)n; MinusNegated ≫ (−u(\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}]))) = (\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}])); SubLessRight ⊳ (−u(\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}]))) = (\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}])) ⊳ ˙¬ ((−u\underline{(ϵ)}) <= (−u(\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}]))) ⇒ ˙¬ ( ˙¬ ((−u\underline{(ϵ)}) = (−u(\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}]))))n)n)n ≫ ˙¬ ((−u\underline{(ϵ)}) <= (\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}])) ⇒ ˙¬ ( ˙¬ ((−u\underline{(ϵ)}) = (\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}])))n)n)n; LessLeq ⊳ ˙¬ ((−u\underline{(ϵ)}) <= (\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}])) ⇒ ˙¬ ( ˙¬ ((−u\underline{(ϵ)}) = (\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}])))n)n)n ≫ (−u\underline{(ϵ)}) <= (\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}])); toNotLess ⊳ (−u\underline{(ϵ)}) <= (\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}])) ≫ ˙¬ ( ˙¬ ((\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}])) <= (−u\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ ((\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}])) = (−u\underline{(ϵ)}))n)n)n)n; NegateDisjunct1 ⊳ ˙¬ ( ˙¬ ((\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}])) <= (−u\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ ((\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}])) = (−u\underline{(ϵ)}))n)n)n)n ⇒ ˙¬ (\underline{(ϵ)} <= (\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}])) ⇒ ˙¬ ( ˙¬ (\underline{(ϵ)} = (\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}])))n)n)n ⊳ ˙¬ ( ˙¬ ((\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}])) <= (−u\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ ((\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}])) = (−u\underline{(ϵ)}))n)n)n)n ≫ ˙¬ (\underline{(ϵ)} <= (\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}])) ⇒ ˙¬ ( ˙¬ (\underline{(ϵ)} = (\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}])))n)n)n; SwitchTerms(x < y − z) ⊳ ˙¬ (\underline{(ϵ)} <= (\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}])) ⇒ ˙¬ ( ˙¬ (\underline{(ϵ)} = (\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}])))n)n)n ≫ ˙¬ (\underline{(fy)}[\underline{m}] <= (\underline{(fx)}[\underline{m}] + (−u\underline{(ϵ)})) ⇒ ˙¬ ( ˙¬ (\underline{(fy)}[\underline{m}] = (\underline{(fx)}[\underline{m}] + (−u\underline{(ϵ)})))n)n)n; LessLeq ⊳ ˙¬ (\underline{(fy)}[\underline{m}] <= (\underline{(fx)}[\underline{m}] + (−u\underline{(ϵ)})) ⇒ ˙¬ ( ˙¬ (\underline{(fy)}[\underline{m}] = (\underline{(fx)}[\underline{m}] + (−u\underline{(ϵ)})))n)n)n ≫ \underline{(fy)}[\underline{m}] <= (\underline{(fx)}[\underline{m}] + (−u\underline{(ϵ)})); ∀\underline{m}: ∀\underline{(n1)}: ∀\underline{(n2)}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{m} ⇒ ˙¬ (\underline{(ϵ)} <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ⇒ ˙¬ ( ˙¬ (\underline{(ϵ)} = |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|)n)n)n)n)n ⊢ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n2)} <= \underline{m} ⇒ ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) = \underline{(ϵ)})n)n)n ⊢ A4 @ \underline{m} ⊳ ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{m} ⇒ ˙¬ (\underline{(ϵ)} <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ⇒ ˙¬ ( ˙¬ (\underline{(ϵ)} = |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|)n)n)n)n)n ≫ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{m} ⇒ \underline{(ϵ)} <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|)n)n; FirstConjunct ⊳ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{m} ⇒ \underline{(ϵ)} <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|)n)n ≫ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n; ∀\underline{m}: ∀\underline{(n1)}: ∀\underline{(n2)}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: Ded ⊳ ∀\underline{m}: ∀\underline{(n1)}: ∀\underline{(n2)}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{m} ⇒ ˙¬ (\underline{(ϵ)} <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ⇒ ˙¬ ( ˙¬ (\underline{(ϵ)} = |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|)n)n)n)n)n ⊢ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n2)} <= \underline{m} ⇒ ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) = \underline{(ϵ)})n)n)n ⊢ if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{m} ⊢ \underline{(fy)}[\underline{m}] <= (\underline{(fx)}[\underline{m}] + (−u\underline{(ϵ)})) ≫ ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{m} ⇒ ˙¬ (\underline{(ϵ)} <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ⇒ ˙¬ ( ˙¬ (\underline{(ϵ)} = |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|)n)n)n)n)n ⇒ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n2)} <= \underline{m} ⇒ ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) = \underline{(ϵ)})n)n)n ⇒ if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{m} ⇒ \underline{(fy)}[\underline{m}] <= (\underline{(fx)}[\underline{m}] + (−u\underline{(ϵ)})); Ded ⊳ ∀\underline{m}: ∀\underline{(n1)}: ∀\underline{(n2)}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{m} ⇒ ˙¬ (\underline{(ϵ)} <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ⇒ ˙¬ ( ˙¬ (\underline{(ϵ)} = |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|)n)n)n)n)n ⊢ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n2)} <= \underline{m} ⇒ ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) = \underline{(ϵ)})n)n)n ⊢ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ≫ ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{m} ⇒ ˙¬ (\underline{(ϵ)} <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ⇒ ˙¬ ( ˙¬ (\underline{(ϵ)} = |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|)n)n)n)n)n ⇒ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n2)} <= \underline{m} ⇒ ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) = \underline{(ϵ)})n)n)n ⇒ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n; JoinConjuncts(2conditions) ⊳ ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{m} ⇒ ˙¬ (\underline{(ϵ)} <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ⇒ ˙¬ ( ˙¬ (\underline{(ϵ)} = |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|)n)n)n)n)n ⇒ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n2)} <= \underline{m} ⇒ ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) = \underline{(ϵ)})n)n)n ⇒ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⊳ ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{m} ⇒ ˙¬ (\underline{(ϵ)} <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ⇒ ˙¬ ( ˙¬ (\underline{(ϵ)} = |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|)n)n)n)n)n ⇒ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n2)} <= \underline{m} ⇒ ˙¬ ((\underline{(fy)}[\underline{m}]+(−u\underline{(fx)}[\underline{m}])) <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}]+(−u\underline{(fx)}[\underline{m}])) = \underline{(ϵ)})n)n)n ⇒ if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{m} ⇒ \underline{(fy)}[\underline{m}] <= (\underline{(fx)}[\underline{m}] + (−u\underline{(ϵ)})) ≫ ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{m} ⇒ ˙¬ (\underline{(ϵ)} <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ⇒ ˙¬ ( ˙¬ (\underline{(ϵ)} = |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|)n)n)n)n)n ⇒ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n2)} <= \underline{m} ⇒ ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) = \underline{(ϵ)})n)n)n ⇒ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{m} ⇒ \underline{(fy)}[\underline{m}] <= (\underline{(fx)}[\underline{m}] + (−u\underline{(ϵ)})))n)n⌉, p_{0}, c)]

[ToLess(F)(Helper) →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{(n1)}: ∀\underline{(n2)}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{m} ⇒ ˙¬ (\underline{(ϵ)} <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ⇒ ˙¬ ( ˙¬ (\underline{(ϵ)} = |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|)n)n)n)n)n ⇒ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n2)} <= \underline{m} ⇒ ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) = \underline{(ϵ)})n)n)n ⇒ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{m} ⇒ \underline{(fy)}[\underline{m}] <= (\underline{(fx)}[\underline{m}] + (−u\underline{(ϵ)})))n)n]

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

[ToLess(F)(Helper) →^{pyk} “lemma toLess(F) helper”]

ToLess(F)

[ToLess(F) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: ∀\underline{(n1)}: ∀\underline{(n2)}: ∀\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)n ⇒ ∀_{obj}\overline{(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}] <= (\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)n ≫ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(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)n; FromNegatedOr ⊳ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(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)n ≫ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(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)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} ⇒ \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)n)n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(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; SecondConjunct ⊳ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(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)n)n)n ≫ ˙¬ (∀_{obj}\overline{(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; FromNot < f(Strong) ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(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 ≫ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n2)} <= \underline{m} ⇒ ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) = \underline{(ϵ)})n)n)n; fromNotSameF(Strongest) ⊳ ˙¬ (∀_{obj}\overline{(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}\underline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\underline{(n1)}: ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{m} ⇒ ˙¬ (\underline{(ϵ)} <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ⇒ ˙¬ ( ˙¬ (\underline{(ϵ)} = |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|)n)n)n)n)n)n)n)n)n; ToLess(F)(Helper) ≫ ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{m} ⇒ ˙¬ (\underline{(ϵ)} <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ⇒ ˙¬ ( ˙¬ (\underline{(ϵ)} = |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|)n)n)n)n)n ⇒ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n2)} <= \underline{m} ⇒ ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) = \underline{(ϵ)})n)n)n ⇒ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{m} ⇒ \underline{(fy)}[\underline{m}] <= (\underline{(fx)}[\underline{m}] + (−u\underline{(ϵ)})))n)n; TwiceExistMP ⊳ ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{m} ⇒ ˙¬ (\underline{(ϵ)} <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ⇒ ˙¬ ( ˙¬ (\underline{(ϵ)} = |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|)n)n)n)n)n ⇒ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n2)} <= \underline{m} ⇒ ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) = \underline{(ϵ)})n)n)n ⇒ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{m} ⇒ \underline{(fy)}[\underline{m}] <= (\underline{(fx)}[\underline{m}] + (−u\underline{(ϵ)})))n)n ⊳ ˙¬ (∀_{obj}\underline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\underline{(n1)}: ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{m} ⇒ ˙¬ (\underline{(ϵ)} <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ⇒ ˙¬ ( ˙¬ (\underline{(ϵ)} = |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|)n)n)n)n)n)n)n)n)n ≫ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n2)} <= \underline{m} ⇒ ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) = \underline{(ϵ)})n)n)n ⇒ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{m} ⇒ \underline{(fy)}[\underline{m}] <= (\underline{(fx)}[\underline{m}] + (−u\underline{(ϵ)})))n)n; MP ⊳ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n2)} <= \underline{m} ⇒ ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) = \underline{(ϵ)})n)n)n ⇒ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{m} ⇒ \underline{(fy)}[\underline{m}] <= (\underline{(fx)}[\underline{m}] + (−u\underline{(ϵ)})))n)n ⊳ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n2)} <= \underline{m} ⇒ ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ ((\underline{(fy)}[\underline{m}] + (−u\underline{(fx)}[\underline{m}])) = \underline{(ϵ)})n)n)n ≫ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{m} ⇒ \underline{(fy)}[\underline{m}] <= (\underline{(fx)}[\underline{m}] + (−u\underline{(ϵ)})))n)n; Gen ⊳ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{m} ⇒ \underline{(fy)}[\underline{m}] <= (\underline{(fx)}[\underline{m}] + (−u\underline{(ϵ)})))n)n ≫ ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{m} ⇒ \underline{(fy)}[\underline{m}] <= (\underline{(fx)}[\underline{m}] + (−u\underline{(ϵ)})))n)n; IntroExist @ if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) ⊳ ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{m} ⇒ \underline{(fy)}[\underline{m}] <= (\underline{(fx)}[\underline{m}] + (−u\underline{(ϵ)})))n)n ≫ ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \underline{(fy)}[\underline{m}] <= (\underline{(fx)}[\underline{m}] + (−u\underline{(ϵ)})))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{(fy)}[\underline{m}] <= (\underline{(fx)}[\underline{m}] + (−u\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{(fy)}[\underline{m}] <= (\underline{(fx)}[\underline{m}] + (−u\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{(fy)}[\underline{m}] <= (\underline{(fx)}[\underline{m}] + (−u\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{(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{(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⌉, p_{0}, c)]

[ToLess(F) →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: ∀\underline{(n1)}: ∀\underline{(n2)}: ∀\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)n ⇒ ∀_{obj}\overline{(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{(fy)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n]

[ToLess(F) →^{tex} “ToLess(F)”]

[ToLess(F) →^{pyk} “lemma toLess(F)”]

FromNot <<

[FromNot << →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: AutoImply ≫ ˙¬ (∀_{obj}\overline{(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; ˙¬ ( ˙¬ (∀_{obj}\overline{(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 ⊢ 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}\overline{(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}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n⌉, p_{0}, c)]

[FromNot << →^{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}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n]

[FromNot <<→^{tex} “FromNot<<”]

[FromNot << →^{pyk} “lemma fromNot<<”]

ToLess(R)

[ToLess(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 ⊢ 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{(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; FromNegatedOr ⊳ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(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{(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)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} ⇒ \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)n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(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; SecondConjunct ⊳ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(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)n)n ≫ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(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; FromNot << ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(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}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)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\})n ≫ ˙¬ (∀_{obj}\overline{(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; JoinConjuncts⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(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)n ≫ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(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)n)n)n; ToNegatedOr ⊳ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(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)n)n)n ≫ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(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)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)n ≫ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(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)n; ToLess(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)n ⇒ ∀_{obj}\overline{(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{(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{(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⌉, p_{0}, c)]

[ToLess(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]

[ToLess(R) →^{tex} “ToLess(R)”]

[ToLess(R) →^{pyk} “lemma toLess(R)”]

FromNotSameF(Weak)(Helper)

[FromNotSameF(Weak)(Helper) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{z} <= |(\underline{x}+(−u\underline{y}))| ⊢ 0 <= (\underline{x}+(−u\underline{y})) ⊢ NonnegativeNumerical⊳0 <= (\underline{x}+(−u\underline{y})) ≫ |(\underline{x} + (−u\underline{y}))| = (\underline{x} + (−u\underline{y})); subLeqRight ⊳ |(\underline{x} + (−u\underline{y}))| = (\underline{x} + (−u\underline{y})) ⊳ \underline{z} <= |(\underline{x} + (−u\underline{y}))| ≫ \underline{z} <= (\underline{x} + (−u\underline{y})); negativeToLeft(Leq) ⊳ \underline{z} <= (\underline{x} + (−u\underline{y})) ≫ (\underline{z} + \underline{y}) <= \underline{x}; plusCommutativity ≫ (\underline{z} + \underline{y}) = (\underline{y} + \underline{z}); subLeqLeft ⊳ (\underline{z} + \underline{y}) = (\underline{y} + \underline{z}) ⊳ (\underline{z} + \underline{y}) <= \underline{x} ≫ (\underline{y} + \underline{z}) <= \underline{x}; PositiveToRight(Leq) ⊳ (\underline{y} + \underline{z}) <= \underline{x} ≫ \underline{y} <= (\underline{x} + (−u\underline{z})); WeakenOr1 ⊳ \underline{y} <= (\underline{x} + (−u\underline{z})) ≫ ˙¬ (\underline{x} <= (\underline{y} + (−u\underline{z})))n ⇒ \underline{y} <= (\underline{x} + (−u\underline{z})); ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{z} <= |(\underline{x} + (−u\underline{y}))| ⊢ ˙¬ (0 <= (\underline{x} + (−u\underline{y})))n ⊢ ToLess ⊳ ˙¬ (0 <= (\underline{x} + (−u\underline{y})))n ≫ ˙¬ ((\underline{x} + (−u\underline{y})) <= 0 ⇒ ˙¬ ( ˙¬ ((\underline{x} + (−u\underline{y})) = 0)n)n)n; NegativeNumerical ≫ |(\underline{x} + (−u\underline{y}))| = (−u(\underline{x} + (−u\underline{y}))); MinusNegated ≫ (−u(\underline{x} + (−u\underline{y}))) = (\underline{y}+(−u\underline{x})); eqTransitivity⊳|(\underline{x}+(−u\underline{y}))| = (−u(\underline{x}+(−u\underline{y})))⊳(−u(\underline{x}+(−u\underline{y}))) = (\underline{y} + (−u\underline{x})) ≫ |(\underline{x} + (−u\underline{y}))| = (\underline{y} + (−u\underline{x})); subLeqRight ⊳ |(\underline{x} + (−u\underline{y}))| = (\underline{y}+(−u\underline{x}))⊳\underline{z} <= |(\underline{x}+(−u\underline{y}))| ≫ \underline{z} <= (\underline{y}+(−u\underline{x})); negativeToLeft(Leq)⊳\underline{z} <= (\underline{y} + (−u\underline{x})) ≫ (\underline{z} + \underline{x}) <= \underline{y}; plusCommutativity ≫ (\underline{z} + \underline{x}) = (\underline{x} + \underline{z}); subLeqLeft ⊳ (\underline{z} + \underline{x}) = (\underline{x} + \underline{z}) ⊳ (\underline{z} + \underline{x}) <= \underline{y} ≫ (\underline{x} + \underline{z}) <= \underline{y}; PositiveToRight(Leq) ⊳ (\underline{x} + \underline{z}) <= \underline{y} ≫ \underline{x} <= (\underline{y} + (−u\underline{z})); WeakenOr2 ⊳ \underline{x} <= (\underline{y} + (−u\underline{z})) ≫ ˙¬ (\underline{x} <= (\underline{y} + (−u\underline{z})))n ⇒ \underline{y} <= (\underline{x} + (−u\underline{z})); ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: Ded ⊳ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{z} <= |(\underline{x} + (−u\underline{y}))| ⊢ 0 <= (\underline{x} + (−u\underline{y})) ⊢ ˙¬ (\underline{x} <= (\underline{y} + (−u\underline{z})))n ⇒ \underline{y} <= (\underline{x} + (−u\underline{z})) ≫ \underline{z} <= |(\underline{x} + (−u\underline{y}))| ⇒ 0 <= (\underline{x} + (−u\underline{y})) ⇒ ˙¬ (\underline{x} <= (\underline{y} + (−u\underline{z})))n ⇒ \underline{y} <= (\underline{x} + (−u\underline{z})); Ded ⊳ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{z} <= |(\underline{x} + (−u\underline{y}))| ⊢ ˙¬ (0 <= (\underline{x} + (−u\underline{y})))n ⊢ ˙¬ (\underline{x} <= (\underline{y} + (−u\underline{z})))n ⇒ \underline{y} <= (\underline{x} + (−u\underline{z})) ≫ \underline{z} <= |(\underline{x} + (−u\underline{y}))| ⇒ ˙¬ (0 <= (\underline{x} + (−u\underline{y})))n ⇒ ˙¬ (\underline{x} <= (\underline{y} + (−u\underline{z})))n ⇒ \underline{y} <= (\underline{x} + (−u\underline{z})); ˙¬ ( ˙¬ (|(\underline{x} + (−u\underline{y}))| <= \underline{z} ⇒ ˙¬ ( ˙¬ (|(\underline{x} + (−u\underline{y}))| = \underline{z})n)n)n)n ⊢ fromNotLess ⊳ ˙¬ ( ˙¬ (|(\underline{x} + (−u\underline{y}))| <= \underline{z} ⇒ ˙¬ ( ˙¬ (|(\underline{x} + (−u\underline{y}))| = \underline{z})n)n)n)n ≫ \underline{z} <= |(\underline{x} + (−u\underline{y}))|; MP ⊳ \underline{z} <= |(\underline{x} + (−u\underline{y}))| ⇒ 0 <= (\underline{x} + (−u\underline{y})) ⇒ ˙¬ (\underline{x} <= (\underline{y} + (−u\underline{z})))n ⇒ \underline{y} <= (\underline{x} + (−u\underline{z})) ⊳ \underline{z} <= |(\underline{x} + (−u\underline{y}))| ≫ 0 <= (\underline{x} + (−u\underline{y})) ⇒ ˙¬ (\underline{x} <= (\underline{y} + (−u\underline{z})))n ⇒ \underline{y} <= (\underline{x} + (−u\underline{z})); MP ⊳ \underline{z} <= |(\underline{x} + (−u\underline{y}))| ⇒ ˙¬ (0 <= (\underline{x} + (−u\underline{y})))n ⇒ ˙¬ (\underline{x} <= (\underline{y} + (−u\underline{z})))n ⇒ \underline{y} <= (\underline{x} + (−u\underline{z})) ⊳ \underline{z} <= |(\underline{x} + (−u\underline{y}))| ≫ ˙¬ (0 <= (\underline{x} + (−u\underline{y})))n ⇒ ˙¬ (\underline{x} <= (\underline{y} + (−u\underline{z})))n ⇒ \underline{y} <= (\underline{x} + (−u\underline{z})); FromNegations ⊳ 0 <= (\underline{x} + (−u\underline{y})) ⇒ ˙¬ (\underline{x} <= (\underline{y} + (−u\underline{z})))n ⇒ \underline{y} <= (\underline{x} + (−u\underline{z})) ⊳ ˙¬ (0 <= (\underline{x} + (−u\underline{y})))n ⇒ ˙¬ (\underline{x} <= (\underline{y} + (−u\underline{z})))n ⇒ \underline{y} <= (\underline{x} + (−u\underline{z})) ≫ ˙¬ (\underline{x} <= (\underline{y} + (−u\underline{z})))n ⇒ \underline{y} <= (\underline{x} + (−u\underline{z}))⌉, p_{0}, c)]

[FromNotSameF(Weak)(Helper) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ ( ˙¬ (|(\underline{x} + (−u\underline{y}))| <= \underline{z} ⇒ ˙¬ ( ˙¬ (|(\underline{x} + (−u\underline{y}))| = \underline{z})n)n)n)n ⊢ ˙¬ (\underline{x} <= (\underline{y} + (−u\underline{z})))n ⇒ \underline{y} <= (\underline{x} + (−u\underline{z}))]

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

[FromNotSameF(Weak)(Helper) →^{pyk} “lemma fromNotSameF(Weak)(Helper)”]

FromNotSameF(Weak)

[FromNotSameF(Weak) →^{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}] + (−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; 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}] + (−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}\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; AEA − negated ⊳ ˙¬ (∀_{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}\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; ∀\underline{m}: ∀\underline{n}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ ( ˙¬ (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 ⊢ FromNegated(2 ∗ Imply) ⊳ ˙¬ ( ˙¬ (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 ≫ ˙¬ ( ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m})n)n ⇒ ˙¬ ( ˙¬ ( ˙¬ (|(\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; FirstConjunct ⊳ ˙¬ ( ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m})n)n ⇒ ˙¬ ( ˙¬ ( ˙¬ (|(\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 ≫ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m})n)n; SecondConjunct ⊳ ˙¬ ( ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m})n)n ⇒ ˙¬ ( ˙¬ ( ˙¬ (|(\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 ≫ ˙¬ ( ˙¬ (|(\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; FromNotSameF(Weak)(Helper)⊳ ˙¬ ( ˙¬ (|(\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 ≫ ˙¬ (\underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})))n ⇒ \underline{(fy)}[\underline{m}] <= (\underline{(fx)}[\underline{m}]+(−u\underline{(ϵ)})); JoinConjuncts⊳ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m})n)n ⊳ ˙¬ (\underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})))n ⇒ \underline{(fy)}[\underline{m}] <= (\underline{(fx)}[\underline{m}] + (−u\underline{(ϵ)})) ≫ ˙¬ ( ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m})n)n ⇒ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})))n ⇒ \underline{(fy)}[\underline{m}] <= (\underline{(fx)}[\underline{m}] + (−u\underline{(ϵ)})))n)n; Ded ⊳ ∀\underline{m}: ∀\underline{n}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ ( ˙¬ (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 ⊢ ˙¬ ( ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m})n)n ⇒ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})))n ⇒ \underline{(fy)}[\underline{m}] <= (\underline{(fx)}[\underline{m}] + (−u\underline{(ϵ)})))n)n ≫ ˙¬ ( ˙¬ (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 ⇒ ˙¬ ( ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m})n)n ⇒ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})))n ⇒ \underline{(fy)}[\underline{m}] <= (\underline{(fx)}[\underline{m}] + (−u\underline{(ϵ)})))n)n; AddEAE ⊳ ˙¬ ( ˙¬ (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 ⇒ ˙¬ ( ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m})n)n ⇒ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})))n ⇒ \underline{(fy)}[\underline{m}] <= (\underline{(fx)}[\underline{m}] + (−u\underline{(ϵ)})))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 ⇒ ˙¬ (∀_{obj}\underline{(ϵ)}: ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m})n)n ⇒ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})))n ⇒ \underline{(fy)}[\underline{m}] <= (\underline{(fx)}[\underline{m}] + (−u\underline{(ϵ)})))n)n)n)n)n)n; MP ⊳ ˙¬ (∀_{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})n)n ⇒ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})))n ⇒ \underline{(fy)}[\underline{m}] <= (\underline{(fx)}[\underline{m}] + (−u\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 ≫ ˙¬ (∀_{obj}\underline{(ϵ)}: ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m})n)n ⇒ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})))n ⇒ \underline{(fy)}[\underline{m}] <= (\underline{(fx)}[\underline{m}] + (−u\underline{(ϵ)})))n)n)n)n)n)n⌉, p_{0}, c)]

[FromNotSameF(Weak) →^{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 ⊢ ˙¬ (∀_{obj}\underline{(ϵ)}: ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m})n)n ⇒ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)})))n ⇒ \underline{(fy)}[\underline{m}] <= (\underline{(fx)}[\underline{m}] + (−u\underline{(ϵ)})))n)n)n)n)n)n]

[FromNotSameF(Weak) →^{tex} “FromNotSameF(Weak)”]

[FromNotSameF(Weak) →^{pyk} “lemma fromNotSameF(Weak)”]

FromNotLess(F)

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

[FromNotLess(F) →^{pyk} “lemma fromNotLess(F)”]

== Addition

[== Addition →^{tex} “==Addition”]

[== Addition →^{pyk} “lemma ==Addition”]

== AdditionLeft

[== AdditionLeft →^{tex} “==AdditionLeft”]

[== AdditionLeft →^{pyk} “lemma ==AdditionLeft”]

Fpart − Bounded(Base)

[Fpart − Bounded(Base) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{(v2n)}: ∀\underline{(fx)}: \underline{(v2n)} <= 0 ⊢ LeqLessEq ⊳ \underline{(v2n)} <= 0 ≫ ˙¬ ( ˙¬ (\underline{(v2n)} <= 0 ⇒ ˙¬ ( ˙¬ (\underline{(v2n)} = 0)n)n)n)n ⇒ \underline{(v2n)} = 0; Nonnegative(N) ≫ 0 <= \underline{(v2n)}; toNotLess ⊳ 0 <= \underline{(v2n)} ≫ ˙¬ ( ˙¬ (\underline{(v2n)} <= 0 ⇒ ˙¬ ( ˙¬ (\underline{(v2n)} = 0)n)n)n)n; NegateDisjunct1 ⊳ ˙¬ ( ˙¬ (\underline{(v2n)} <= 0 ⇒ ˙¬ ( ˙¬ (\underline{(v2n)} = 0)n)n)n)n ⇒ \underline{(v2n)} = 0 ⊳ ˙¬ ( ˙¬ (\underline{(v2n)} <= 0 ⇒ ˙¬ ( ˙¬ (\underline{(v2n)} = 0)n)n)n)n ≫ \underline{(v2n)} = 0; SameSeries⊳\underline{(v2n)} = 0 ≫ \underline{(fx)}[\underline{(v2n)}] = \underline{(fx)}[0]; SameNumerical⊳\underline{(fx)}[\underline{(v2n)}] = \underline{(fx)}[0] ≫ |\underline{(fx)}[\underline{(v2n)}]| = |\underline{(fx)}[0]|; eqAddition ⊳ |\underline{(fx)}[\underline{(v2n)}]| = |\underline{(fx)}[0]| ≫ (|\underline{(fx)}[\underline{(v2n)}]| + 1) = (|\underline{(fx)}[0]| + 1); leqReflexivity ≫ |\underline{(fx)}[\underline{(v2n)}]| <= |\underline{(fx)}[\underline{(v2n)}]|; Leq + 1 ⊳ |\underline{(fx)}[\underline{(v2n)}]| <= |\underline{(fx)}[\underline{(v2n)}]| ≫ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= (|\underline{(fx)}[\underline{(v2n)}]| + 1) ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = (|\underline{(fx)}[\underline{(v2n)}]| + 1))n)n)n; SubLessRight ⊳ (|\underline{(fx)}[\underline{(v2n)}]| + 1) = (|\underline{(fx)}[0]| + 1) ⊳ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= (|\underline{(fx)}[\underline{(v2n)}]| + 1) ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = (|\underline{(fx)}[\underline{(v2n)}]| + 1))n)n)n ≫ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= (|\underline{(fx)}[0]| + 1) ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = (|\underline{(fx)}[0]| + 1))n)n)n; ∀\underline{(v1)}: ∀\underline{(v2n)}: ∀\underline{(fx)}: Ded ⊳ ∀\underline{(v1)}: ∀\underline{(v2n)}: ∀\underline{(fx)}: \underline{(v2n)} <= 0 ⊢ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= (|\underline{(fx)}[0]| + 1) ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = (|\underline{(fx)}[0]| + 1))n)n)n ≫ \underline{(v2n)} <= 0 ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= (|\underline{(fx)}[0]| + 1) ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = (|\underline{(fx)}[0]| + 1))n)n)n; Gen ⊳ \underline{(v2n)} <= 0 ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= (|\underline{(fx)}[0]| + 1) ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = (|\underline{(fx)}[0]| + 1))n)n)n ≫ ∀_{obj}\underline{(v2n)}: \underline{(v2n)} <= 0 ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= (|\underline{(fx)}[0]| + 1) ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = (|\underline{(fx)}[0]| + 1))n)n)n; IntroExist @(|\underline{(fx)}[0]| + 1) ⊳ ∀_{obj}\underline{(v2n)}: \underline{(v2n)} <= 0 ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= (|\underline{(fx)}[0]| + 1) ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = (|\underline{(fx)}[0]| + 1))n)n)n ≫ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (∀_{obj}\underline{(v2n)}: \underline{(v2n)} <= 0 ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = \underline{(v1)})n)n)n)n)n⌉, p_{0}, c)]

[Fpart − Bounded(Base) →^{stmt} SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{(v2n)}: ∀\underline{(fx)}: ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (∀_{obj}\underline{(v2n)}: \underline{(v2n)} <= 0 ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = \underline{(v1)})n)n)n)n)n]

[Fpart − Bounded(Base) →^{tex} “Fpart-Bounded(Base)”] [Fpart − Bounded(Base) →^{pyk} “lemma fpart-Bounded base”]

Fpart − Bounded(InduHelper)

[Fpart − Bounded(InduHelper) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{(v2n)}: ∀\underline{n}: ∀\underline{(fx)}: \underline{(v2n)} <= \underline{n} ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = \underline{(v1)})n)n)n ⊢ \underline{(v2n)} <= (\underline{n} + 1) ⊢ LeqLessEq ⊳ \underline{(v2n)} <= (\underline{n} + 1) ≫ ˙¬ ( ˙¬ (\underline{(v2n)} <= (\underline{n} + 1) ⇒ ˙¬ ( ˙¬ (\underline{(v2n)} = (\underline{n} + 1))n)n)n)n ⇒ \underline{(v2n)} = (\underline{n} + 1); ∀\underline{(v1)}: ∀\underline{(v2n)}: ∀\underline{n}: ∀\underline{(fx)}: \underline{(v2n)} <= \underline{n} ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = \underline{(v1)})n)n)n ⊢ ˙¬ (\underline{(v2n)} <= (\underline{n} + 1) ⇒ ˙¬ ( ˙¬ (\underline{(v2n)} = (\underline{n} + 1))n)n)n ⊢ LessMinus1(N) ⊳ ˙¬ (\underline{(v2n)} <= (\underline{n} + 1) ⇒ ˙¬ ( ˙¬ (\underline{(v2n)} = (\underline{n} + 1))n)n)n ≫ \underline{(v2n)} <= \underline{n}; MP ⊳ \underline{(v2n)} <= \underline{n} ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = \underline{(v1)})n)n)n ⊳ \underline{(v2n)} <= \underline{n} ≫ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = \underline{(v1)})n)n)n; MaxLeq(1) ≫ \underline{(v1)} <= if((|\underline{(fx)}[(\underline{n}+1)]|+1) <= \underline{(v1)}, \underline{(v1)}, (|\underline{(fx)}[(\underline{n} + 1)]| + 1)); LessLeqTransitivity ⊳ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = \underline{(v1)})n)n)n ⊳ \underline{(v1)} <= if((|\underline{(fx)}[(\underline{n} + 1)]| + 1) <= \underline{(v1)}, \underline{(v1)}, (|\underline{(fx)}[(\underline{n} + 1)]| + 1)) ≫ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= if((|\underline{(fx)}[(\underline{n} + 1)]| + 1) <= \underline{(v1)}, \underline{(v1)}, (|\underline{(fx)}[(\underline{n} + 1)]| + 1)) ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = if((|\underline{(fx)}[(\underline{n} + 1)]| + 1) <= \underline{(v1)}, \underline{(v1)}, (|\underline{(fx)}[(\underline{n} + 1)]| + 1)))n)n)n; ∀\underline{(v1)}: ∀\underline{(v2n)}: ∀\underline{n}: ∀\underline{(fx)}: \underline{(v2n)} = (\underline{n} + 1) ⊢ SameSeries ⊳ \underline{(v2n)} = (\underline{n} + 1) ≫ \underline{(fx)}[\underline{(v2n)}] = \underline{(fx)}[(\underline{n} + 1)]; SameNumerical ⊳ \underline{(fx)}[\underline{(v2n)}] = \underline{(fx)}[(\underline{n} + 1)] ≫ |\underline{(fx)}[\underline{(v2n)}]| = |\underline{(fx)}[(\underline{n} + 1)]|; eqLeq ⊳ |\underline{(fx)}[\underline{(v2n)}]| = |\underline{(fx)}[(\underline{n} + 1)]| ≫ |\underline{(fx)}[\underline{(v2n)}]| <= |\underline{(fx)}[(\underline{n} + 1)]|; Leq + 1 ⊳ |\underline{(fx)}[\underline{(v2n)}]| <= |\underline{(fx)}[(\underline{n} + 1)]| ≫ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= (|\underline{(fx)}[(\underline{n}+1)]|+1) ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = (|\underline{(fx)}[(\underline{n}+1)]|+1))n)n)n; MaxLeq(2) ≫ (|\underline{(fx)}[(\underline{n} + 1)]| + 1) <= if((|\underline{(fx)}[(\underline{n} + 1)]| + 1) <= \underline{(v1)}, \underline{(v1)}, (|\underline{(fx)}[(\underline{n} + 1)]| + 1)); LessLeqTransitivity ⊳ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= (|\underline{(fx)}[(\underline{n} + 1)]| + 1) ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = (|\underline{(fx)}[(\underline{n} + 1)]| + 1))n)n)n ⊳ (|\underline{(fx)}[(\underline{n} + 1)]| + 1) <= if((|\underline{(fx)}[(\underline{n} + 1)]| + 1) <= \underline{(v1)}, \underline{(v1)}, (|\underline{(fx)}[(\underline{n} + 1)]| + 1)) ≫ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= if((|\underline{(fx)}[(\underline{n} + 1)]| + 1) <= \underline{(v1)}, \underline{(v1)}, (|\underline{(fx)}[(\underline{n} + 1)]| + 1)) ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = if((|\underline{(fx)}[(\underline{n} + 1)]| + 1) <= \underline{(v1)}, \underline{(v1)}, (|\underline{(fx)}[(\underline{n} + 1)]| + 1)))n)n)n; Ded ⊳ ∀\underline{(v1)}: ∀\underline{(v2n)}: ∀\underline{n}: ∀\underline{(fx)}: \underline{(v2n)} <= \underline{n} ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = \underline{(v1)})n)n)n ⊢ ˙¬ (\underline{(v2n)} <= (\underline{n} + 1) ⇒ ˙¬ ( ˙¬ (\underline{(v2n)} = (\underline{n} + 1))n)n)n ⊢ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= if((|\underline{(fx)}[(\underline{n} + 1)]| + 1) <= \underline{(v1)}, \underline{(v1)}, (|\underline{(fx)}[(\underline{n} + 1)]| + 1)) ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = if((|\underline{(fx)}[(\underline{n} + 1)]| + 1) <= \underline{(v1)}, \underline{(v1)}, (|\underline{(fx)}[(\underline{n} + 1)]| + 1)))n)n)n ≫ \underline{(v2n)} <= \underline{n} ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = \underline{(v1)})n)n)n ⇒ ˙¬ (\underline{(v2n)} <= (\underline{n} + 1) ⇒ ˙¬ ( ˙¬ (\underline{(v2n)} = (\underline{n} + 1))n)n)n ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= if((|\underline{(fx)}[(\underline{n} + 1)]| + 1) <= \underline{(v1)}, \underline{(v1)}, (|\underline{(fx)}[(\underline{n} + 1)]| + 1)) ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = if((|\underline{(fx)}[(\underline{n} + 1)]| + 1) <= \underline{(v1)}, \underline{(v1)}, (|\underline{(fx)}[(\underline{n} + 1)]| + 1)))n)n)n; MP ⊳ \underline{(v2n)} <= \underline{n} ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = \underline{(v1)})n)n)n ⇒ ˙¬ (\underline{(v2n)} <= (\underline{n} + 1) ⇒ ˙¬ ( ˙¬ (\underline{(v2n)} = (\underline{n} + 1))n)n)n ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= if((|\underline{(fx)}[(\underline{n} + 1)]| + 1) <= \underline{(v1)}, \underline{(v1)}, (|\underline{(fx)}[(\underline{n} + 1)]| + 1)) ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = if((|\underline{(fx)}[(\underline{n} + 1)]| + 1) <= \underline{(v1)}, \underline{(v1)}, (|\underline{(fx)}[(\underline{n} + 1)]| + 1)))n)n)n ⊳ \underline{(v2n)} <= \underline{n} ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = \underline{(v1)})n)n)n ≫ ˙¬ (\underline{(v2n)} <= (\underline{n} + 1) ⇒ ˙¬ ( ˙¬ (\underline{(v2n)} = (\underline{n} + 1))n)n)n ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= if((|\underline{(fx)}[(\underline{n} + 1)]| + 1) <= \underline{(v1)}, \underline{(v1)}, (|\underline{(fx)}[(\underline{n} + 1)]| + 1)) ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = if((|\underline{(fx)}[(\underline{n} + 1)]| + 1) <= \underline{(v1)}, \underline{(v1)}, (|\underline{(fx)}[(\underline{n} + 1)]| + 1)))n)n)n; Ded ⊳ ∀\underline{(v1)}: ∀\underline{(v2n)}: ∀\underline{n}: ∀\underline{(fx)}: \underline{(v2n)} = (\underline{n} + 1) ⊢ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= if((|\underline{(fx)}[(\underline{n} + 1)]| + 1) <= \underline{(v1)}, \underline{(v1)}, (|\underline{(fx)}[(\underline{n} + 1)]| + 1)) ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = if((|\underline{(fx)}[(\underline{n} + 1)]| + 1) <= \underline{(v1)}, \underline{(v1)}, (|\underline{(fx)}[(\underline{n} + 1)]| + 1)))n)n)n ≫ \underline{(v2n)} = (\underline{n} + 1) ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= if((|\underline{(fx)}[(\underline{n} + 1)]| + 1) <= \underline{(v1)}, \underline{(v1)}, (|\underline{(fx)}[(\underline{n} + 1)]| + 1)) ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = if((|\underline{(fx)}[(\underline{n} + 1)]| + 1) <= \underline{(v1)}, \underline{(v1)}, (|\underline{(fx)}[(\underline{n} + 1)]| + 1)))n)n)n; FromDisjuncts ⊳ ˙¬ ( ˙¬ (\underline{(v2n)} <= (\underline{n} + 1) ⇒ ˙¬ ( ˙¬ (\underline{(v2n)} = (\underline{n} + 1))n)n)n)n ⇒ \underline{(v2n)} = (\underline{n} + 1) ⊳ ˙¬ (\underline{(v2n)} <= (\underline{n} + 1) ⇒ ˙¬ ( ˙¬ (\underline{(v2n)} = (\underline{n} + 1))n)n)n ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= if((|\underline{(fx)}[(\underline{n} + 1)]| + 1) <= \underline{(v1)}, \underline{(v1)}, (|\underline{(fx)}[(\underline{n} + 1)]| + 1)) ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = if((|\underline{(fx)}[(\underline{n} + 1)]| + 1) <= \underline{(v1)}, \underline{(v1)}, (|\underline{(fx)}[(\underline{n} + 1)]| + 1)))n)n)n ⊳ \underline{(v2n)} = (\underline{n} + 1) ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= if((|\underline{(fx)}[(\underline{n} + 1)]| + 1) <= \underline{(v1)}, \underline{(v1)}, (|\underline{(fx)}[(\underline{n} + 1)]| + 1)) ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = if((|\underline{(fx)}[(\underline{n} + 1)]| + 1) <= \underline{(v1)}, \underline{(v1)}, (|\underline{(fx)}[(\underline{n} + 1)]| + 1)))n)n)n ≫ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= if((|\underline{(fx)}[(\underline{n} + 1)]| + 1) <= \underline{(v1)}, \underline{(v1)}, (|\underline{(fx)}[(\underline{n} + 1)]| + 1)) ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = if((|\underline{(fx)}[(\underline{n} + 1)]| + 1) <= \underline{(v1)}, \underline{(v1)}, (|\underline{(fx)}[(\underline{n} + 1)]| + 1)))n)n)n⌉, p_{0}, c)]

[Fpart − Bounded(InduHelper) →^{stmt} SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{(v2n)}: ∀\underline{n}: ∀\underline{(fx)}: \underline{(v2n)} <= \underline{n} ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = \underline{(v1)})n)n)n ⊢ \underline{(v2n)} <= (\underline{n} + 1) ⊢ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= if((|\underline{(fx)}[(\underline{n} + 1)]| + 1) <= \underline{(v1)}, \underline{(v1)}, (|\underline{(fx)}[(\underline{n} + 1)]| + 1)) ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = if((|\underline{(fx)}[(\underline{n} + 1)]| + 1) <= \underline{(v1)}, \underline{(v1)}, (|\underline{(fx)}[(\underline{n} + 1)]| + 1)))n)n)n]

[Fpart − Bounded(InduHelper) →^{tex} “Fpart-Bounded(InduHelper)”]

[Fpart − Bounded(InduHelper) →^{pyk} “lemma fpart-Bounded indu helper”]

Fpart − Bounded(Indu)

[Fpart − Bounded(Indu) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{(v2n)}: ∀\underline{n}: ∀\underline{(fx)}: \underline{(v2n)} <= \underline{n} ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = \underline{(v1)})n)n)n ⊢ \underline{(v2n)} <= (\underline{n} + 1) ⊢ Fpart − Bounded(InduHelper) ⊳ \underline{(v2n)} <= \underline{n} ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = \underline{(v1)})n)n)n ⊳ \underline{(v2n)} <= (\underline{n} + 1) ≫ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= if((|\underline{(fx)}[(\underline{n} + 1)]| + 1) <= \underline{(v1)}, \underline{(v1)}, (|\underline{(fx)}[(\underline{n} + 1)]| + 1)) ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = if((|\underline{(fx)}[(\underline{n} + 1)]| + 1) <= \underline{(v1)}, \underline{(v1)}, (|\underline{(fx)}[(\underline{n} + 1)]| + 1)))n)n)n; ∀\underline{(v1)}: ∀\underline{(v2n)}: ∀\underline{n}: ∀\underline{(fx)}: Ded ⊳ ∀\underline{(v1)}: ∀\underline{(v2n)}: ∀\underline{n}: ∀\underline{(fx)}: \underline{(v2n)} <= \underline{n} ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = \underline{(v1)})n)n)n ⊢ \underline{(v2n)} <= (\underline{n}+1) ⊢ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= if((|\underline{(fx)}[(\underline{n} + 1)]| + 1) <= \underline{(v1)}, \underline{(v1)}, (|\underline{(fx)}[(\underline{n} + 1)]| + 1)) ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = if((|\underline{(fx)}[(\underline{n} + 1)]| + 1) <= \underline{(v1)}, \underline{(v1)}, (|\underline{(fx)}[(\underline{n} + 1)]| + 1)))n)n)n ≫ \underline{(v2n)} <= \underline{n} ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = \underline{(v1)})n)n)n ⇒ \underline{(v2n)} <= (\underline{n} + 1) ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= if((|\underline{(fx)}[(\underline{n} + 1)]| + 1) <= \underline{(v1)}, \underline{(v1)}, (|\underline{(fx)}[(\underline{n} + 1)]| + 1)) ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = if((|\underline{(fx)}[(\underline{n} + 1)]| + 1) <= \underline{(v1)}, \underline{(v1)}, (|\underline{(fx)}[(\underline{n} + 1)]| + 1)))n)n)n; AddAll ⊳ \underline{(v2n)} <= \underline{n} ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = \underline{(v1)})n)n)n ⇒ \underline{(v2n)} <= (\underline{n} + 1) ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= if((|\underline{(fx)}[(\underline{n} + 1)]| + 1) <= \underline{(v1)}, \underline{(v1)}, (|\underline{(fx)}[(\underline{n} + 1)]| + 1)) ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = if((|\underline{(fx)}[(\underline{n} + 1)]| + 1) <= \underline{(v1)}, \underline{(v1)}, (|\underline{(fx)}[(\underline{n} + 1)]| + 1)))n)n)n ≫ ∀_{obj}\underline{(v2n)}: \underline{(v2n)} <= \underline{n} ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = \underline{(v1)})n)n)n ⇒ ∀_{obj}\underline{(v2n)}: \underline{(v2n)} <= (\underline{n} + 1) ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= if((|\underline{(fx)}[(\underline{n} + 1)]| + 1) <= \underline{(v1)}, \underline{(v1)}, (|\underline{(fx)}[(\underline{n} + 1)]| + 1)) ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = if((|\underline{(fx)}[(\underline{n} + 1)]| + 1) <= \underline{(v1)}, \underline{(v1)}, (|\underline{(fx)}[(\underline{n} + 1)]| + 1)))n)n)n; AddExist(SimpleAnt) @ if((|\underline{(fx)}[(\underline{n} + 1)]| + 1) <= \underline{(v1)}, \underline{(v1)}, (|\underline{(fx)}[(\underline{n} + 1)]| + 1)) ⊳ ∀_{obj}\underline{(v2n)}: \underline{(v2n)} <= \underline{n} ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = \underline{(v1)})n)n)n ⇒ ∀_{obj}\underline{(v2n)}: \underline{(v2n)} <= (\underline{n} + 1) ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= if((|\underline{(fx)}[(\underline{n} + 1)]| + 1) <= \underline{(v1)}, \underline{(v1)}, (|\underline{(fx)}[(\underline{n} + 1)]| + 1)) ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = if((|\underline{(fx)}[(\underline{n} + 1)]| + 1) <= \underline{(v1)}, \underline{(v1)}, (|\underline{(fx)}[(\underline{n} + 1)]| + 1)))n)n)n ≫ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (∀_{obj}\underline{(v2n)}: \underline{(v2n)} <= \underline{n} ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = \underline{(v1)})n)n)n)n)n ⇒ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (∀_{obj}\underline{(v2n)}: \underline{(v2n)} <= (\underline{n} + 1) ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = \underline{(v1)})n)n)n)n)n⌉, p_{0}, c)]

[Fpart − Bounded(Indu) →^{stmt} SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{(v2n)}: ∀\underline{n}: ∀\underline{(fx)}: ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (∀_{obj}\underline{(v2n)}: \underline{(v2n)} <= \underline{n} ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = \underline{(v1)})n)n)n)n)n ⇒ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (∀_{obj}\underline{(v2n)}: \underline{(v2n)} <= (\underline{n} + 1) ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2n)}]| = \underline{(v1)})n)n)n)n)n]

[Fpart − Bounded(Indu) →^{tex} “Fpart-Bounded(Indu)”]

[Fpart − Bounded(Indu) →^{pyk} “lemma fpart-Bounded indu”]

Fpart − Bounded

[Fpart − Bounded →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{n}: ∀\underline{(fx)}: Fpart − Bounded(Base) ≫ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (∀_{obj}\underline{(v2)}: \underline{(v2)} <= 0 ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2)}]| = \underline{(v1)})n)n)n)n)n; Fpart − Bounded(Indu) ≫ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (∀_{obj}\underline{(v2)}: \underline{(v2)} <= \underline{n} ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2)}]| = \underline{(v1)})n)n)n)n)n ⇒ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (∀_{obj}\underline{(v2)}: \underline{(v2)} <= (\underline{n} + 1) ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2)}]| = \underline{(v1)})n)n)n)n)n; Induction ⊳ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (∀_{obj}\underline{(v2)}: \underline{(v2)} <= 0 ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2)}]| = \underline{(v1)})n)n)n)n)n ⊳ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (∀_{obj}\underline{(v2)}: \underline{(v2)} <= \underline{n} ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2)}]| = \underline{(v1)})n)n)n)n)n ⇒ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (∀_{obj}\underline{(v2)}: \underline{(v2)} <= (\underline{n} + 1) ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2)}]| = \underline{(v1)})n)n)n)n)n ≫ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (∀_{obj}\underline{(v2)}: \underline{(v2)} <= \underline{n} ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2)}]| = \underline{(v1)})n)n)n)n)n⌉, p_{0}, c)]

[Fpart − Bounded →^{stmt} SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{n}: ∀\underline{(fx)}: ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (∀_{obj}\underline{(v2)}: \underline{(v2)} <= \underline{n} ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2)}]| = \underline{(v1)})n)n)n)n)n]

[Fpart − Bounded →^{tex} “Fpart-Bounded”]

[Fpart − Bounded →^{pyk} “lemma fpart-Bounded”]

F − Bounded(Helper)

[F − Bounded(Helper) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ (|(\underline{x} + (−u\underline{y}))| <= \underline{z} ⇒ ˙¬ ( ˙¬ (|(\underline{x} + (−u\underline{y}))| = \underline{z})n)n)n ⊢ NumericalDifferenceLess ⊳ ˙¬ (|(\underline{x} + (−u\underline{y}))| <= \underline{z} ⇒ ˙¬ ( ˙¬ (|(\underline{x} + (−u\underline{y}))| = \underline{z})n)n)n ≫ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) <= \underline{x} ⇒ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) = \underline{x})n)n)n ⇒ ˙¬ ( ˙¬ (\underline{x} <= (\underline{y} + \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = (\underline{y} + \underline{z}))n)n)n)n)n; FirstConjunct ⊳ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) <= \underline{x} ⇒ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) = \underline{x})n)n)n ⇒ ˙¬ ( ˙¬ (\underline{x} <= (\underline{y} + \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = (\underline{y} + \underline{z}))n)n)n)n)n ≫ ˙¬ ((\underline{y} + (−u\underline{z})) <= \underline{x} ⇒ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) = \underline{x})n)n)n; NegativeToRight(Less) ⊳ ˙¬ ((\underline{y} + (−u\underline{z})) <= \underline{x} ⇒ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) = \underline{x})n)n)n ≫ ˙¬ (\underline{y} <= (\underline{x} + \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{y} = (\underline{x} + \underline{z}))n)n)n; x <= |x| ≫ (\underline{x} + \underline{z}) <= |(\underline{x} + \underline{z})|; LessLeqTransitivity ⊳ ˙¬ (\underline{y} <= (\underline{x} + \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{y} = (\underline{x} + \underline{z}))n)n)n ⊳ (\underline{x} + \underline{z}) <= |(\underline{x} + \underline{z})| ≫ ˙¬ (\underline{y} <= |(\underline{x} + \underline{z})| ⇒ ˙¬ ( ˙¬ (\underline{y} = |(\underline{x} + \underline{z})|)n)n)n; MaxLeq(1) ≫ |(\underline{x} + \underline{z})| <= if(|(\underline{x} + (−u\underline{z}))| <= |(\underline{x} + \underline{z})|, |(\underline{x} + \underline{z})|, |(\underline{x} + (−u\underline{z}))|); LessLeqTransitivity ⊳ ˙¬ (\underline{y} <= |(\underline{x} + \underline{z})| ⇒ ˙¬ ( ˙¬ (\underline{y} = |(\underline{x} + \underline{z})|)n)n)n ⊳ |(\underline{x} + \underline{z})| <= if(|(\underline{x} + (−u\underline{z}))| <= |(\underline{x} + \underline{z})|, |(\underline{x} + \underline{z})|, |(\underline{x} + (−u\underline{z}))|) ≫ ˙¬ (\underline{y} <= if(|(\underline{x} + (−u\underline{z}))| <= |(\underline{x} + \underline{z})|, |(\underline{x} + \underline{z})|, |(\underline{x} + (−u\underline{z}))|) ⇒ ˙¬ ( ˙¬ (\underline{y} = if(|(\underline{x} + (−u\underline{z}))| <= |(\underline{x} + \underline{z})|, |(\underline{x} + \underline{z})|, |(\underline{x} + (−u\underline{z}))|))n)n)n; SecondConjunct ⊳ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) <= \underline{x} ⇒ ˙¬ ( ˙¬ ((\underline{y} + (−u\underline{z})) = \underline{x})n)n)n ⇒ ˙¬ ( ˙¬ (\underline{x} <= (\underline{y} + \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = (\underline{y} + \underline{z}))n)n)n)n)n ≫ ˙¬ (\underline{x} <= (\underline{y} + \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = (\underline{y} + \underline{z}))n)n)n; PositiveToLeft(Less) ⊳ ˙¬ (\underline{x} <= (\underline{y} + \underline{z}) ⇒ ˙¬ ( ˙¬ (\underline{x} = (\underline{y} + \underline{z}))n)n)n ≫ ˙¬ ((\underline{x} + (−u\underline{z})) <= \underline{y} ⇒ ˙¬ ( ˙¬ ((\underline{x} + (−u\underline{z})) = \underline{y})n)n)n; x <= |x| ≫ (\underline{z} + (−u\underline{x})) <= |(\underline{z} + (−u\underline{x}))|; NumericalDifference ≫ |(\underline{z} + (−u\underline{x}))| = |(\underline{x} + (−u\underline{z}))|; subLeqRight ⊳ |(\underline{z} + (−u\underline{x}))| = |(\underline{x} + (−u\underline{z}))| ⊳ (\underline{z} + (−u\underline{x})) <= |(\underline{z} + (−u\underline{x}))| ≫ (\underline{z} + (−u\underline{x})) <= |(\underline{x} + (−u\underline{z}))|; LeqNegated ⊳ (\underline{z} + (−u\underline{x})) <= |(\underline{x} + (−u\underline{z}))| ≫ (−u|(\underline{x} + (−u\underline{z}))|) <= (−u(\underline{z} + (−u\underline{x}))); MinusNegated ≫ (−u(\underline{z} + (−u\underline{x}))) = (\underline{x} + (−u\underline{z})); subLeqRight ⊳ (−u(\underline{z} + (−u\underline{x}))) = (\underline{x} + (−u\underline{z})) ⊳ (−u|(\underline{x} + (−u\underline{z}))|) <= (−u(\underline{z} + (−u\underline{x}))) ≫ (−u|(\underline{x} + (−u\underline{z}))|) <= (\underline{x} + (−u\underline{z})); leqLessTransitivity ⊳ (−u|(\underline{x} + (−u\underline{z}))|) <= (\underline{x} + (−u\underline{z})) ⊳ ˙¬ ((\underline{x} + (−u\underline{z})) <= \underline{y} ⇒ ˙¬ ( ˙¬ ((\underline{x} + (−u\underline{z})) = \underline{y})n)n)n ≫ ˙¬ ((−u|(\underline{x} + (−u\underline{z}))|) <= \underline{y} ⇒ ˙¬ ( ˙¬ ((−u|(\underline{x} + (−u\underline{z}))|) = \underline{y})n)n)n; MaxLeq(2) ≫ |(\underline{x} + (−u\underline{z}))| <= if(|(\underline{x}+(−u\underline{z}))| <= |(\underline{x}+\underline{z})|, |(\underline{x}+\underline{z})|, |(\underline{x}+(−u\underline{z}))|); LeqNegated⊳|(\underline{x}+(−u\underline{z}))| <= if(|(\underline{x} + (−u\underline{z}))| <= |(\underline{x} + \underline{z})|, |(\underline{x} + \underline{z})|, |(\underline{x} + (−u\underline{z}))|) ≫ (−uif(|(\underline{x} + (−u\underline{z}))| <= |(\underline{x} + \underline{z})|, |(\underline{x} + \underline{z})|, |(\underline{x} + (−u\underline{z}))|)) <= (−u|(\underline{x} + (−u\underline{z}))|); leqLessTransitivity ⊳ (−uif(|(\underline{x} + (−u\underline{z}))| <= |(\underline{x} + \underline{z})|, |(\underline{x} + \underline{z})|, |(\underline{x} + (−u\underline{z}))|)) <= (−u|(\underline{x} + (−u\underline{z}))|) ⊳ ˙¬ ((−u|(\underline{x} + (−u\underline{z}))|) <= \underline{y} ⇒ ˙¬ ( ˙¬ ((−u|(\underline{x} + (−u\underline{z}))|) = \underline{y})n)n)n ≫ ˙¬ ((−uif(|(\underline{x} + (−u\underline{z}))| <= |(\underline{x} + \underline{z})|, |(\underline{x} + \underline{z})|, |(\underline{x} + (−u\underline{z}))|)) <= \underline{y} ⇒ ˙¬ ( ˙¬ ((−uif(|(\underline{x} + (−u\underline{z}))| <= |(\underline{x} + \underline{z})|, |(\underline{x} + \underline{z})|, |(\underline{x} + (−u\underline{z}))|)) = \underline{y})n)n)n; ToNumericalLess ⊳ ˙¬ ((−uif(|(\underline{x} + (−u\underline{z}))| <= |(\underline{x} + \underline{z})|, |(\underline{x} + \underline{z})|, |(\underline{x} + (−u\underline{z}))|)) <= \underline{y} ⇒ ˙¬ ( ˙¬ ((−uif(|(\underline{x}+(−u\underline{z}))| <= |(\underline{x}+\underline{z})|, |(\underline{x}+\underline{z})|, |(\underline{x}+(−u\underline{z}))|)) = \underline{y})n)n)n ⊳ ˙¬ (\underline{y} <= if(|(\underline{x} + (−u\underline{z}))| <= |(\underline{x} + \underline{z})|, |(\underline{x} + \underline{z})|, |(\underline{x} + (−u\underline{z}))|) ⇒ ˙¬ ( ˙¬ (\underline{y} = if(|(\underline{x} + (−u\underline{z}))| <= |(\underline{x} + \underline{z})|, |(\underline{x} + \underline{z})|, |(\underline{x} + (−u\underline{z}))|))n)n)n ≫ ˙¬ (|\underline{y}| <= if(|(\underline{x} + (−u\underline{z}))| <= |(\underline{x} + \underline{z})|, |(\underline{x} + \underline{z})|, |(\underline{x} + (−u\underline{z}))|) ⇒ ˙¬ ( ˙¬ (|\underline{y}| = if(|(\underline{x} + (−u\underline{z}))| <= |(\underline{x} + \underline{z})|, |(\underline{x} + \underline{z})|, |(\underline{x} + (−u\underline{z}))|))n)n)n⌉, p_{0}, c)]

[F − Bounded(Helper) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ (|(\underline{x} + (−u\underline{y}))| <= \underline{z} ⇒ ˙¬ ( ˙¬ (|(\underline{x} + (−u\underline{y}))| = \underline{z})n)n)n ⊢ ˙¬ (|\underline{y}| <= if(|(\underline{x} + (−u\underline{z}))| <= |(\underline{x} + \underline{z})|, |(\underline{x} + \underline{z})|, |(\underline{x} + (−u\underline{z}))|) ⇒ ˙¬ ( ˙¬ (|\underline{y}| = if(|(\underline{x} + (−u\underline{z}))| <= |(\underline{x} + \underline{z})|, |(\underline{x} + \underline{z})|, |(\underline{x} + (−u\underline{z}))|))n)n)n]

[F − Bounded(Helper) →^{tex} “F-Bounded(Helper)”]

[F − Bounded(Helper) →^{pyk} “lemma f-Bounded helper”]

F − Bounded

[F − Bounded →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{n}: ∀\underline{(fx)}: ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= 1 ⇒ ˙¬ ( ˙¬ (0 = 1)n)n)n ⇒ \underline{n} <= \underline{(v1)} ⇒ \underline{n} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= 1 ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = 1)n)n)n ⊢ \underline{n} <= \underline{(v2)} ⊢ A4 @ \underline{n} ⊳ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= 1 ⇒ ˙¬ ( ˙¬ (0 = 1)n)n)n ⇒ \underline{n} <= \underline{(v1)} ⇒ \underline{n} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= 1 ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = 1)n)n)n ≫ ∀_{obj}\underline{(v2)}: ˙¬ (0 <= 1 ⇒ ˙¬ ( ˙¬ (0 = 1)n)n)n ⇒ \underline{n} <= \underline{n} ⇒ \underline{n} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{n}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= 1 ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{n}] + (−u\underline{(fx)}[\underline{(v2)}]))| = 1)n)n)n; A4 @ \underline{(v2)} ⊳ ∀_{obj}\underline{(v2)}: ˙¬ (0 <= 1 ⇒ ˙¬ ( ˙¬ (0 = 1)n)n)n ⇒ \underline{n} <= \underline{n} ⇒ \underline{n} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{n}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= 1 ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{n}] + (−u\underline{(fx)}[\underline{(v2)}]))| = 1)n)n)n ≫ ˙¬ (0 <= 1 ⇒ ˙¬ ( ˙¬ (0 = 1)n)n)n ⇒ \underline{n} <= \underline{n} ⇒ \underline{n} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{n}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= 1 ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{n}] + (−u\underline{(fx)}[\underline{(v2)}]))| = 1)n)n)n; 0 < 1 ≫ ˙¬ (0 <= 1 ⇒ ˙¬ ( ˙¬ (0 = 1)n)n)n; leqReflexivity ≫ \underline{n} <= \underline{n}; MP3 ⊳ ˙¬ (0 <= 1 ⇒ ˙¬ ( ˙¬ (0 = 1)n)n)n ⇒ \underline{n} <= \underline{n} ⇒ \underline{n} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{n}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= 1 ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{n}] + (−u\underline{(fx)}[\underline{(v2)}]))| = 1)n)n)n ⊳ ˙¬ (0 <= 1 ⇒ ˙¬ ( ˙¬ (0 = 1)n)n)n ⊳ \underline{n} <= \underline{n} ⊳ \underline{n} <= \underline{(v2)} ≫ ˙¬ (|(\underline{(fx)}[\underline{n}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= 1 ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{n}] + (−u\underline{(fx)}[\underline{(v2)}]))| = 1)n)n)n; F − Bounded(Helper) ⊳ ˙¬ (|(\underline{(fx)}[\underline{n}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= 1 ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{n}] + (−u\underline{(fx)}[\underline{(v2)}]))| = 1)n)n)n ≫ ˙¬ (|\underline{(fx)}[\underline{(v2)}]| <= if(|(\underline{(fx)}[\underline{n}] + (−u1))| <= |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + (−u1))|) ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2)}]| = if(|(\underline{(fx)}[\underline{n}] + (−u1))| <= |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + (−u1))|))n)n)n; ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{n}: ∀\underline{(fx)}: ∀_{obj}\underline{(v2)}: \underline{(v2)} <= \underline{n} ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2)}]| = \underline{(v1)})n)n)n ⊢ ˙¬ (\underline{n} <= \underline{(v2)})n ⊢ A4 @ \underline{(v2)} ⊳ ∀_{obj}\underline{(v2)}: \underline{(v2)} <= \underline{n} ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2)}]| = \underline{(v1)})n)n)n ≫ \underline{(v2)} <= \underline{n} ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2)}]| = \underline{(v1)})n)n)n; ToLess ⊳ ˙¬ (\underline{n} <= \underline{(v2)})n ≫ ˙¬ (\underline{(v2)} <= \underline{n} ⇒ ˙¬ ( ˙¬ (\underline{(v2)} = \underline{n})n)n)n; LessLeq ⊳ ˙¬ (\underline{(v2)} <= \underline{n} ⇒ ˙¬ ( ˙¬ (\underline{(v2)} = \underline{n})n)n)n ≫ \underline{(v2)} <= \underline{n}; MP ⊳ \underline{(v2)} <= \underline{n} ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2)}]| = \underline{(v1)})n)n)n ⊳ \underline{(v2)} <= \underline{n} ≫ ˙¬ (|\underline{(fx)}[\underline{(v2)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2)}]| = \underline{(v1)})n)n)n; ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{n}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: Ded ⊳ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{n}: ∀\underline{(fx)}: ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= 1 ⇒ ˙¬ ( ˙¬ (0 = 1)n)n)n ⇒ \underline{n} <= \underline{(v1)} ⇒ \underline{n} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= 1 ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = 1)n)n)n ⊢ \underline{n} <= \underline{(v2)} ⊢ ˙¬ (|\underline{(fx)}[\underline{(v2)}]| <= if(|(\underline{(fx)}[\underline{n}] + (−u1))| <= |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + (−u1))|) ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2)}]| = if(|(\underline{(fx)}[\underline{n}] + (−u1))| <= |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + (−u1))|))n)n)n ≫ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= 1 ⇒ ˙¬ ( ˙¬ (0 = 1)n)n)n ⇒ \underline{n} <= \underline{(v1)} ⇒ \underline{n} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= 1 ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}]+(−u\underline{(fx)}[\underline{(v2)}]))| = 1)n)n)n ⇒ \underline{n} <= \underline{(v2)} ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2)}]| <= if(|(\underline{(fx)}[\underline{n}] + (−u1))| <= |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + (−u1))|) ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2)}]| = if(|(\underline{(fx)}[\underline{n}] + (−u1))| <= |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + (−u1))|))n)n)n; Ded ⊳ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{n}: ∀\underline{(fx)}: ∀_{obj}\underline{(v2)}: \underline{(v2)} <= \underline{n} ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2)}]| = \underline{(v1)})n)n)n ⊢ ˙¬ (\underline{n} <= \underline{(v2)})n ⊢ ˙¬ (|\underline{(fx)}[\underline{(v2)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2)}]| = \underline{(v1)})n)n)n ≫ ∀_{obj}\underline{(v2)}: \underline{(v2)} <= \underline{n} ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2)}]| = \underline{(v1)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{(v2)})n ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2)}]| = \underline{(v1)})n)n)n; Cauchy ≫ ∀_{obj}\underline{(ϵ)}: ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{n} <= \underline{(v1)} ⇒ \underline{n} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n)n)n; A4 @ 1 ⊳ ∀_{obj}\underline{(ϵ)}: ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{n} <= \underline{(v1)} ⇒ \underline{n} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n)n)n ≫ ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= 1 ⇒ ˙¬ ( ˙¬ (0 = 1)n)n)n ⇒ \underline{n} <= \underline{(v1)} ⇒ \underline{n} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= 1 ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = 1)n)n)n)n)n; Fpart − Bounded ≫ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (∀_{obj}\underline{(v2)}: \underline{(v2)} <= \underline{n} ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2)}]| = \underline{(v1)})n)n)n)n)n; ExistMP ⊳ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= 1 ⇒ ˙¬ ( ˙¬ (0 = 1)n)n)n ⇒ \underline{n} <= \underline{(v1)} ⇒ \underline{n} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= 1 ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = 1)n)n)n ⇒ \underline{n} <= \underline{(v2)} ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2)}]| <= if(|(\underline{(fx)}[\underline{n}] + (−u1))| <= |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + (−u1))|) ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2)}]| = if(|(\underline{(fx)}[\underline{n}]+(−u1))| <= |(\underline{(fx)}[\underline{n}]+1)|, |(\underline{(fx)}[\underline{n}]+1)|, |(\underline{(fx)}[\underline{n}]+(−u1))|))n)n)n⊳ ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= 1 ⇒ ˙¬ ( ˙¬ (0 = 1)n)n)n ⇒ \underline{n} <= \underline{(v1)} ⇒ \underline{n} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= 1 ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = 1)n)n)n)n)n ≫ \underline{n} <= \underline{(v2)} ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2)}]| <= if(|(\underline{(fx)}[\underline{n}] + (−u1))| <= |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + (−u1))|) ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2)}]| = if(|(\underline{(fx)}[\underline{n}]+(−u1))| <= |(\underline{(fx)}[\underline{n}]+1)|, |(\underline{(fx)}[\underline{n}]+1)|, |(\underline{(fx)}[\underline{n}]+ (−u1))|))n)n)n; ExistMP ⊳ ∀_{obj}\underline{(v2)}: \underline{(v2)} <= \underline{n} ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2)}]| = \underline{(v1)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{(v2)})n ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2)}]| = \underline{(v1)})n)n)n ⊳ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (∀_{obj}\underline{(v2)}: \underline{(v2)} <= \underline{n} ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2)}]| = \underline{(v1)})n)n)n)n)n ≫ ˙¬ (\underline{n} <= \underline{(v2)})n ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2)}]| = \underline{(v1)})n)n)n; LessThanMax ⊳ \underline{n} <= \underline{(v2)} ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2)}]| <= if(|(\underline{(fx)}[\underline{n}] + (−u1))| <= |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + (−u1))|) ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2)}]| = if(|(\underline{(fx)}[\underline{n}] + (−u1))| <= |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + (−u1))|))n)n)n ⊳ ˙¬ (\underline{n} <= \underline{(v2)})n ⇒ ˙¬ (|\underline{(fx)}[\underline{(v2)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2)}]| = \underline{(v1)})n)n)n ≫ ˙¬ (|\underline{(fx)}[\underline{(v2)}]| <= if(\underline{(v1)} <= if(|(\underline{(fx)}[\underline{n}] + (−u1))| <= |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + (−u1))|), if(|(\underline{(fx)}[\underline{n}] + (−u1))| <= |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + (−u1))|), \underline{(v1)}) ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2)}]| = if(\underline{(v1)} <= if(|(\underline{(fx)}[\underline{n}] + (−u1))| <= |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + (−u1))|), if(|(\underline{(fx)}[\underline{n}] + (−u1))| <= |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + (−u1))|), \underline{(v1)}))n)n)n; Gen ⊳ ˙¬ (|\underline{(fx)}[\underline{(v2)}]| <= if(\underline{(v1)} <= if(|(\underline{(fx)}[\underline{n}] + (−u1))| <= |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + (−u1))|), if(|(\underline{(fx)}[\underline{n}] + (−u1))| <= |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + (−u1))|), \underline{(v1)}) ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2)}]| = if(\underline{(v1)} <= if(|(\underline{(fx)}[\underline{n}] + (−u1))| <= |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + (−u1))|), if(|(\underline{(fx)}[\underline{n}] + (−u1))| <= |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + (−u1))|), \underline{(v1)}))n)n)n ≫ ∀_{obj}\underline{(v2)}: ˙¬ (|\underline{(fx)}[\underline{(v2)}]| <= if(\underline{(v1)} <= if(|(\underline{(fx)}[\underline{n}] + (−u1))| <= |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + (−u1))|), if(|(\underline{(fx)}[\underline{n}] + (−u1))| <= |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + (−u1))|), \underline{(v1)}) ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2)}]| = if(\underline{(v1)} <= if(|(\underline{(fx)}[\underline{n}] + (−u1))| <= |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + (−u1))|), if(|(\underline{(fx)}[\underline{n}] + (−u1))| <= |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + (−u1))|), \underline{(v1)}))n)n)n; IntroExist @ if(\underline{(v1)} <= if(|(\underline{(fx)}[\underline{n}] + (−u1))| <= |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + (−u1))|), if(|(\underline{(fx)}[\underline{n}] + (−u1))| <= |(\underline{(fx)}[\underline{n}]+1)|, |(\underline{(fx)}[\underline{n}]+1)|, |(\underline{(fx)}[\underline{n}]+(−u1))|), \underline{(v1)})⊳∀_{obj}\underline{(v2)}: ˙¬ (|\underline{(fx)}[\underline{(v2)}]| <= if(\underline{(v1)} <= if(|(\underline{(fx)}[\underline{n}] + (−u1))| <= |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + (−u1))|), if(|(\underline{(fx)}[\underline{n}] + (−u1))| <= |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + (−u1))|), \underline{(v1)}) ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2)}]| = if(\underline{(v1)} <= if(|(\underline{(fx)}[\underline{n}] + (−u1))| <= |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + (−u1))|), if(|(\underline{(fx)}[\underline{n}] + (−u1))| <= |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + 1)|, |(\underline{(fx)}[\underline{n}] + (−u1))|), \underline{(v1)}))n)n)n ≫ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (|\underline{(fx)}[\underline{(v2)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2)}]| = \underline{(v1)})n)n)n)n)n⌉, p_{0}, c)]

[F − Bounded →^{stmt} SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{n}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (|\underline{(fx)}[\underline{(v2)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fx)}[\underline{(v2)}]| = \underline{(v1)})n)n)n)n)n]

[F − Bounded →^{tex} “F-Bounded”]

[F − Bounded →^{pyk} “lemma f-Bounded”]

SameFmultiplication(Helper)

[SameFmultiplication(Helper) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{m}: ∀\underline{n}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀_{obj}\underline{m}: ˙¬ (0 <= (\underline{(ϵ)} ∗ rec\underline{(v1)}) ⇒ ˙¬ ( ˙¬ (0 = (\underline{(ϵ)} ∗ rec\underline{(v1)}))n)n)n ⇒ \underline{n} <= \underline{m} ⇒ ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| <= (\underline{(ϵ)} ∗ rec\underline{(v1)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| = (\underline{(ϵ)} ∗ rec\underline{(v1)}))n)n)n ⊢ ∀_{obj}\underline{(v2)}: ˙¬ (|\underline{(fz)}[\underline{(v2)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fz)}[\underline{(v2)}]| = \underline{(v1)})n)n)n ⊢ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⊢ \underline{n} <= \underline{m} ⊢ A4 @ \underline{(v2)} ⊳ ∀_{obj}\underline{(v2)}: ˙¬ (|\underline{(fz)}[\underline{(v2)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fz)}[\underline{(v2)}]| = \underline{(v1)})n)n)n ≫ ˙¬ (|\underline{(fz)}[\underline{(v2)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fz)}[\underline{(v2)}]| = \underline{(v1)})n)n)n; 0 <= |x| ≫ 0 <= |\underline{(fz)}[\underline{(v2)}]|; leqLessTransitivity ⊳ 0 <= |\underline{(fz)}[\underline{(v2)}]| ⊳ ˙¬ (|\underline{(fz)}[\underline{(v2)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fz)}[\underline{(v2)}]| = \underline{(v1)})n)n)n ≫ ˙¬ (0 <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(v1)})n)n)n; PositiveInverted ⊳ ˙¬ (0 <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(v1)})n)n)n ≫ ˙¬ (0 <= rec\underline{(v1)} ⇒ ˙¬ ( ˙¬ (0 = rec\underline{(v1)})n)n)n; PositiveFactors ⊳ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⊳ ˙¬ (0 <= rec\underline{(v1)} ⇒ ˙¬ ( ˙¬ (0 = rec\underline{(v1)})n)n)n ≫ ˙¬ (0 <= (\underline{(ϵ)} ∗ rec\underline{(v1)}) ⇒ ˙¬ ( ˙¬ (0 = (\underline{(ϵ)} ∗ rec\underline{(v1)}))n)n)n; A4 @ \underline{m} ⊳ ∀_{obj}\underline{m}: ˙¬ (0 <= (\underline{(ϵ)} ∗ rec\underline{(v1)}) ⇒ ˙¬ ( ˙¬ (0 = (\underline{(ϵ)} ∗ rec\underline{(v1)}))n)n)n ⇒ \underline{n} <= \underline{m} ⇒ ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| <= (\underline{(ϵ)} ∗ rec\underline{(v1)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| = (\underline{(ϵ)} ∗ rec\underline{(v1)}))n)n)n ≫ ˙¬ (0 <= (\underline{(ϵ)} ∗ rec\underline{(v1)}) ⇒ ˙¬ ( ˙¬ (0 = (\underline{(ϵ)} ∗ rec\underline{(v1)}))n)n)n ⇒ \underline{n} <= \underline{m} ⇒ ˙¬ (|(\underline{(fx)}[\underline{m}]+(−u\underline{(fy)}[\underline{m}]))| <= (\underline{(ϵ)}∗rec\underline{(v1)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}]+(−u\underline{(fy)}[\underline{m}]))| = (\underline{(ϵ)} ∗ rec\underline{(v1)}))n)n)n; MP2 ⊳ ˙¬ (0 <= (\underline{(ϵ)} ∗ rec\underline{(v1)}) ⇒ ˙¬ ( ˙¬ (0 = (\underline{(ϵ)} ∗ rec\underline{(v1)}))n)n)n ⇒ \underline{n} <= \underline{m} ⇒ ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| <= (\underline{(ϵ)}∗rec\underline{(v1)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}]+(−u\underline{(fy)}[\underline{m}]))| = (\underline{(ϵ)}∗rec\underline{(v1)}))n)n)n⊳ ˙¬ (0 <= (\underline{(ϵ)} ∗ rec\underline{(v1)}) ⇒ ˙¬ ( ˙¬ (0 = (\underline{(ϵ)} ∗ rec\underline{(v1)}))n)n)n ⊳ \underline{n} <= \underline{m} ≫ ˙¬ (|(\underline{(fx)}[\underline{m}]+(−u\underline{(fy)}[\underline{m}]))| <= (\underline{(ϵ)}∗rec\underline{(v1)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}]+(−u\underline{(fy)}[\underline{m}]))| = (\underline{(ϵ)} ∗ rec\underline{(v1)}))n)n)n; reciprocalToLeft(Less) ⊳ ˙¬ (0 <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(v1)})n)n)n ⊳ ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| <= (\underline{(ϵ)} ∗ rec\underline{(v1)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| = (\underline{(ϵ)} ∗ rec\underline{(v1)}))n)n)n ≫ ˙¬ ((|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ∗ \underline{(v1)}) <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ ((|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ∗ \underline{(v1)}) = \underline{(ϵ)})n)n)n; TimesF ≫ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] = (\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}]); TimesF ≫ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}] = (\underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}]); EqNegated ⊳ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}] = (\underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}]) ≫ (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]) = (−u(\underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])); −x ∗ y = −(x ∗ y) ≫ ((−u\underline{(fy)}[\underline{m}]) ∗ \underline{(fz)}[\underline{m}]) = (−u(\underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])); eqSymmetry ⊳ ((−u\underline{(fy)}[\underline{m}]) ∗ \underline{(fz)}[\underline{m}]) = (−u(\underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])) ≫ (−u(\underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])) = ((−u\underline{(fy)}[\underline{m}]) ∗ \underline{(fz)}[\underline{m}]); eqTransitivity ⊳ (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]) = (−u(\underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])) ⊳ (−u(\underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])) = ((−u\underline{(fy)}[\underline{m}]) ∗ \underline{(fz)}[\underline{m}]) ≫ (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]) = ((−u\underline{(fy)}[\underline{m}]) ∗ \underline{(fz)}[\underline{m}]); AddEquations ⊳ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] = (\underline{(fx)}[\underline{m}]∗\underline{(fz)}[\underline{m}])⊳(−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]) = ((−u\underline{(fy)}[\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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])) = ((\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}]) + ((−u\underline{(fy)}[\underline{m}]) ∗ \underline{(fz)}[\underline{m}])); DistributionOutLeft ≫ ((\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}]) + ((−u\underline{(fy)}[\underline{m}]) ∗ \underline{(fz)}[\underline{m}])) = (\underline{(fz)}[\underline{m}] ∗ (\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))); eqTransitivity ⊳ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])) = ((\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}]) + ((−u\underline{(fy)}[\underline{m}]) ∗ \underline{(fz)}[\underline{m}])) ⊳ ((\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}]) + ((−u\underline{(fy)}[\underline{m}]) ∗ \underline{(fz)}[\underline{m}])) = (\underline{(fz)}[\underline{m}] ∗ (\underline{(fx)}[\underline{m}] + (−u\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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])) = (\underline{(fz)}[\underline{m}] ∗ (\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))); SameNumerical ⊳ (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}])) = (\underline{(fz)}[\underline{m}] ∗ (\underline{(fx)}[\underline{m}] + (−u\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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]))| = |(\underline{(fz)}[\underline{m}] ∗ (\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}])))|; SplitNumericalProduct ≫ |(\underline{(fz)}[\underline{m}] ∗ (\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}])))| = (|\underline{(fz)}[\underline{m}]| ∗ |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|); timesCommutativity ≫ (|\underline{(fz)}[\underline{m}]| ∗ |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|) = (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ∗ |\underline{(fz)}[\underline{m}]|); eqTransitivity4 ⊳ |(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]))| = |(\underline{(fz)}[\underline{m}] ∗ (\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}])))| ⊳ |(\underline{(fz)}[\underline{m}] ∗ (\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}])))| = (|\underline{(fz)}[\underline{m}]| ∗ |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|) ⊳ (|\underline{(fz)}[\underline{m}]| ∗ |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|) = (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]))| = (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ∗ |\underline{(fz)}[\underline{m}]|); eqSymmetry ⊳ |(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]))| = (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|∗|\underline{(fz)}[\underline{m}]|) ≫ (|(\underline{(fx)}[\underline{m}]+(−u\underline{(fy)}[\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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]))|; A4 @ \underline{(fz)}[\underline{m}] ⊳ ∀_{obj}\underline{(v2)}: ˙¬ (|\underline{(fz)}[\underline{(v2)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fz)}[\underline{(v2)}]| = \underline{(v1)})n)n)n ≫ |\underline{(fz)}[\underline{m}]| <= \underline{(v1)}; 0 <= |x| ≫ 0 <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|; LeqMultiplicationLeft ⊳ 0 <= |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ⊳ |\underline{(fz)}[\underline{m}]| <= \underline{(v1)} ≫ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ∗ |\underline{(fz)}[\underline{m}]|) <= (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ∗ \underline{(v1)}); leqLessTransitivity ⊳ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ∗ |\underline{(fz)}[\underline{m}]|) <= (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ∗ \underline{(v1)}) ⊳ ˙¬ ((|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ∗ \underline{(v1)}) <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ ((|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ∗ \underline{(v1)}) = \underline{(ϵ)})n)n)n ≫ ˙¬ ((|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ∗ |\underline{(fz)}[\underline{m}]|) <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ ((|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ∗ |\underline{(fz)}[\underline{m}]|) = \underline{(ϵ)})n)n)n; SubLessLeft ⊳ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]))| ⊳ ˙¬ ((|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ∗ |\underline{(fz)}[\underline{m}]|) <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ ((|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ∗ |\underline{(fz)}[\underline{m}]|) = \underline{(ϵ)})n)n)n ≫ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]))| = \underline{(ϵ)})n)n)n; ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{m}: ∀\underline{n}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: Ded ⊳ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{m}: ∀\underline{n}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀_{obj}\underline{m}: ˙¬ (0 <= (\underline{(ϵ)} ∗ rec\underline{(v1)}) ⇒ ˙¬ ( ˙¬ (0 = (\underline{(ϵ)} ∗ rec\underline{(v1)}))n)n)n ⇒ \underline{n} <= \underline{m} ⇒ ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| <= (\underline{(ϵ)} ∗ rec\underline{(v1)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| = (\underline{(ϵ)} ∗ rec\underline{(v1)}))n)n)n ⊢ ∀_{obj}\underline{(v2)}: ˙¬ (|\underline{(fz)}[\underline{(v2)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fz)}[\underline{(v2)}]| = \underline{(v1)})n)n)n ⊢ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⊢ \underline{n} <= \underline{m} ⊢ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]))| = \underline{(ϵ)})n)n)n ≫ ∀_{obj}\underline{m}: ˙¬ (0 <= (\underline{(ϵ)} ∗ rec\underline{(v1)}) ⇒ ˙¬ ( ˙¬ (0 = (\underline{(ϵ)} ∗ rec\underline{(v1)}))n)n)n ⇒ \underline{n} <= \underline{m} ⇒ ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| <= (\underline{(ϵ)} ∗ rec\underline{(v1)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| = (\underline{(ϵ)} ∗ rec\underline{(v1)}))n)n)n ⇒ ∀_{obj}\underline{(v2)}: ˙¬ (|\underline{(fz)}[\underline{(v2)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fz)}[\underline{(v2)}]| = \underline{(v1)})n)n)n ⇒ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{n} <= \underline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]))| = \underline{(ϵ)})n)n)n⌉, p_{0}, c)]

[SameFmultiplication(Helper) →^{stmt} SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{m}: ∀\underline{n}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀_{obj}\underline{m}: ˙¬ (0 <= (\underline{(ϵ)} ∗ rec\underline{(v1)}) ⇒ ˙¬ ( ˙¬ (0 = (\underline{(ϵ)} ∗ rec\underline{(v1)}))n)n)n ⇒ \underline{n} <= \underline{m} ⇒ ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| <= (\underline{(ϵ)} ∗ rec\underline{(v1)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| = (\underline{(ϵ)} ∗ rec\underline{(v1)}))n)n)n ⇒ ∀_{obj}\underline{(v2)}: ˙¬ (|\underline{(fz)}[\underline{(v2)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fz)}[\underline{(v2)}]| = \underline{(v1)})n)n)n ⇒ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{n} <= \underline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]))| = \underline{(ϵ)})n)n)n]

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

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

SameFmultiplication

[SameFmultiplication →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{m}: ∀\underline{n}: ∀\underline{(ϵ)}: ∀\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}] + (−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}] + (−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; 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}] + (−u\underline{(fy)}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = \overline{(ϵ)})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; A4 @(\underline{(ϵ)} ∗ rec\underline{(v1)}) ⊳ ∀_{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 ≫ ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{m}: ˙¬ (0 <= (\underline{(ϵ)} ∗ rec\underline{(v1)}) ⇒ ˙¬ ( ˙¬ (0 = (\underline{(ϵ)} ∗ rec\underline{(v1)}))n)n)n ⇒ \underline{n} <= \underline{m} ⇒ ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| <= (\underline{(ϵ)} ∗ rec\underline{(v1)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| = (\underline{(ϵ)} ∗ rec\underline{(v1)}))n)n)n)n)n; F − Bounded ≫ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (|\underline{(fz)}[\underline{(v2)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fz)}[\underline{(v2)}]| = \underline{(v1)})n)n)n)n)n; SameFmultiplication(Helper) ≫ ∀_{obj}\underline{m}: ˙¬ (0 <= (\underline{(ϵ)} ∗ rec\underline{(v1)}) ⇒ ˙¬ ( ˙¬ (0 = (\underline{(ϵ)} ∗ rec\underline{(v1)}))n)n)n ⇒ \underline{n} <= \underline{m} ⇒ ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| <= (\underline{(ϵ)} ∗ rec\underline{(v1)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| = (\underline{(ϵ)} ∗ rec\underline{(v1)}))n)n)n ⇒ ∀_{obj}\underline{(v2)}: ˙¬ (|\underline{(fz)}[\underline{(v2)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fz)}[\underline{(v2)}]| = \underline{(v1)})n)n)n ⇒ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{n} <= \underline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]))| = \underline{(ϵ)})n)n)n; ExistMP2 ⊳ ∀_{obj}\underline{m}: ˙¬ (0 <= (\underline{(ϵ)} ∗ rec\underline{(v1)}) ⇒ ˙¬ ( ˙¬ (0 = (\underline{(ϵ)} ∗ rec\underline{(v1)}))n)n)n ⇒ \underline{n} <= \underline{m} ⇒ ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| <= (\underline{(ϵ)} ∗ rec\underline{(v1)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| = (\underline{(ϵ)} ∗ rec\underline{(v1)}))n)n)n ⇒ ∀_{obj}\underline{(v2)}: ˙¬ (|\underline{(fz)}[\underline{(v2)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fz)}[\underline{(v2)}]| = \underline{(v1)})n)n)n ⇒ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{n} <= \underline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]))| = \underline{(ϵ)})n)n)n ⊳ ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{m}: ˙¬ (0 <= (\underline{(ϵ)} ∗ rec\underline{(v1)}) ⇒ ˙¬ ( ˙¬ (0 = (\underline{(ϵ)} ∗ rec\underline{(v1)}))n)n)n ⇒ \underline{n} <= \underline{m} ⇒ ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| <= (\underline{(ϵ)} ∗ rec\underline{(v1)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| = (\underline{(ϵ)} ∗ rec\underline{(v1)}))n)n)n)n)n ⊳ ˙¬ (∀_{obj}\underline{(v1)}: ˙¬ (∀_{obj}\underline{(v2)}: ˙¬ (|\underline{(fz)}[\underline{(v2)}]| <= \underline{(v1)} ⇒ ˙¬ ( ˙¬ (|\underline{(fz)}[\underline{(v2)}]| = \underline{(v1)})n)n)n)n)n ≫ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{n} <= \underline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]))| = \underline{(ϵ)})n)n)n; Gen ⊳ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{n} <= \underline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]))| = \underline{(ϵ)})n)n)n ≫ ∀_{obj}\underline{m}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{n} <= \underline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]))| = \underline{(ϵ)})n)n)n; IntroExist @ \underline{n} ⊳ ∀_{obj}\underline{m}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{n} <= \underline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]))| = \underline{(ϵ)})n)n)n ≫ ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{m}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{n} <= \underline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]))| = \underline{(ϵ)})n)n)n)n)n; Gen ⊳ ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{m}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{n} <= \underline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]))| = \underline{(ϵ)})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} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]))| = \underline{(ϵ)})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} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]))| = \underline{(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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}[\overline{m}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\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} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}[\overline{m}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}[\overline{m}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}[\overline{m}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n⌉, p_{0}, c)]

[SameFmultiplication →^{stmt} SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{m}: ∀\underline{n}: ∀\underline{(ϵ)}: ∀\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}] + (−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} ⇒ ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}[\overline{m}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}[\overline{m}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}]∗\underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n]

[SameFmultiplication →^{tex} “SameFmultiplication”]

[SameFmultiplication →^{pyk} “lemma sameFmultiplication”]

EqMultiplication(R)

[EqMultiplication(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\} ⊢ 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; SameFmultiplication ⊳ ∀_{obj}\overline{(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} ⇒ ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}[\overline{m}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}[\overline{m}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\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} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}[\overline{m}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}[\overline{m}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(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}: ˙¬ (e_{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\}; eqReflexivity ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}; eqReflexivity ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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\}; 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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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\}⌉, p_{0}, c)]

[EqMultiplication(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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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\}]

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

[EqMultiplication(R) →^{pyk} “lemma eqMultiplication(R)”]

EqMultiplicationLeft(R)

[EqMultiplicationLeft(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\} ⊢ EqMultiplication(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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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\}; 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}, (\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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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\}; 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}, (\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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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\}; ==Transitivity ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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\}; ==Transitivity ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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)]

[EqMultiplicationLeft(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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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\}]

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

[EqMultiplicationLeft(R) →^{pyk} “lemma eqMultiplicationLeft(R)”]

x ∗ 0 = 0(F)

[x ∗ 0 = 0(F) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{(fx)}: TimesF ≫ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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}]); NatType ≫ \underline{m} ∈ N; 0f ⊳ \underline{m} ∈ N ≫ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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}] = 0; EqMultiplicationLeft ⊳ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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}] = 0 ≫ (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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}] ∗ 0); x ∗ 0 = 0 ≫ (\underline{(fx)}[\underline{m}] ∗ 0) = 0; eqSymmetry ⊳ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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}] = 0 ≫ 0 = \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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}]; eqTransitivity5 ⊳ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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}] ∗ 0) ⊳ (\underline{(fx)}[\underline{m}] ∗ 0) = 0 ⊳ 0 = {ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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}: ˙¬ (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}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}]; Gen⊳\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}] ≫ ∀_{obj}\underline{m}: \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\underline{m}]; To = f ⊳ ∀_{obj}\underline{m}: \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}\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}: ˙¬ (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\} = \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}⌉, p_{0}, c)]

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

[x ∗ 0 = 0(F) →^{tex} “x∗0=0(F)”]

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

x ∗ 0 = 0(R)

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

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

LessMultiplication(F)(Helper2)

[LessMultiplication(F)(Helper2) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: ∀\underline{v}: ˙¬ (0 <= \underline{u} ⇒ ˙¬ ( ˙¬ (0 = \underline{u})n)n)n ⊢ ˙¬ (0 <= \underline{v} ⇒ ˙¬ ( ˙¬ (0 = \underline{v})n)n)n ⊢ \underline{x} <= (\underline{y} + (−u\underline{u})) ⊢ 0 <= (\underline{z} + (−u\underline{v})) ⊢ negativeToLeft(Leq) ⊳ \underline{x} <= (\underline{y} + (−u\underline{u})) ≫ (\underline{x} + \underline{u}) <= \underline{y}; plusCommutativity ≫ (\underline{x} + \underline{u}) = (\underline{u} + \underline{x}); subLeqLeft ⊳ (\underline{x} + \underline{u}) = (\underline{u} + \underline{x}) ⊳ (\underline{x} + \underline{u}) <= \underline{y} ≫ (\underline{u} + \underline{x}) <= \underline{y}; PositiveToRight(Leq) ⊳ (\underline{u} + \underline{x}) <= \underline{y} ≫ \underline{u} <= (\underline{y} + (−u\underline{x})); negativeToLeft(Leq)(1term) ⊳ 0 <= (\underline{z} + (−u\underline{v})) ≫ \underline{v} <= \underline{z}; LessLeq ⊳ ˙¬ (0 <= \underline{u} ⇒ ˙¬ ( ˙¬ (0 = \underline{u})n)n)n ≫ 0 <= \underline{u}; LessLeq ⊳ ˙¬ (0 <= \underline{v} ⇒ ˙¬ ( ˙¬ (0 = \underline{v})n)n)n ≫ 0 <= \underline{v}; MultiplyEquations(Leq)⊳0 <= \underline{u}⊳0 <= \underline{v}⊳\underline{u} <= (\underline{y} + (−u\underline{x})) ⊳ \underline{v} <= \underline{z} ≫ (\underline{u} ∗ \underline{v}) <= ((\underline{y} + (−u\underline{x})) ∗ \underline{z}); timesCommutativity ≫ ((\underline{y} + (−u\underline{x})) ∗ \underline{z}) = (\underline{z} ∗ (\underline{y} + (−u\underline{x}))); DistributionLeft ≫ (\underline{z} ∗ (\underline{y} + (−u\underline{x}))) = ((\underline{y} ∗ \underline{z}) + ((−u\underline{x}) ∗ \underline{z})); −x ∗ y = −(x ∗ y) ≫ ((−u\underline{x}) ∗ \underline{z}) = (−u(\underline{x}∗\underline{z})); EqAdditionLeft⊳((−u\underline{x})∗\underline{z}) = (−u(\underline{x}∗\underline{z})) ≫ ((\underline{y}∗\underline{z})+((−u\underline{x})∗\underline{z})) = ((\underline{y} ∗ \underline{z}) + (−u(\underline{x} ∗ \underline{z}))); eqTransitivity4 ⊳ ((\underline{y} + (−u\underline{x})) ∗ \underline{z}) = (\underline{z}∗(\underline{y}+(−u\underline{x})))⊳(\underline{z}∗(\underline{y}+(−u\underline{x}))) = ((\underline{y}∗\underline{z})+((−u\underline{x})∗\underline{z}))⊳((\underline{y}∗\underline{z})+((−u\underline{x})∗\underline{z})) = ((\underline{y} ∗ \underline{z}) + (−u(\underline{x} ∗ \underline{z}))) ≫ ((\underline{y} + (−u\underline{x})) ∗ \underline{z}) = ((\underline{y} ∗ \underline{z}) + (−u(\underline{x} ∗ \underline{z}))); subLeqRight ⊳ ((\underline{y}+(−u\underline{x}))∗\underline{z}) = ((\underline{y}∗\underline{z})+(−u(\underline{x}∗\underline{z})))⊳(\underline{u}∗\underline{v}) <= ((\underline{y}+(−u\underline{x}))∗\underline{z}) ≫ (\underline{u}∗\underline{v}) <= ((\underline{y} ∗ \underline{z}) + (−u(\underline{x} ∗ \underline{z}))); negativeToLeft(Leq) ⊳ (\underline{u} ∗ \underline{v}) <= ((\underline{y} ∗ \underline{z}) + (−u(\underline{x} ∗ \underline{z}))) ≫ ((\underline{u} ∗ \underline{v}) + (\underline{x} ∗ \underline{z})) <= (\underline{y} ∗ \underline{z}); plusCommutativity ≫ ((\underline{u} ∗ \underline{v}) + (\underline{x} ∗ \underline{z})) = ((\underline{x}∗\underline{z})+(\underline{u}∗\underline{v})); subLeqLeft⊳((\underline{u}∗\underline{v})+(\underline{x}∗\underline{z})) = ((\underline{x}∗\underline{z})+(\underline{u}∗\underline{v}))⊳((\underline{u}∗\underline{v})+(\underline{x}∗\underline{z})) <= (\underline{y}∗\underline{z}) ≫ ((\underline{x}∗\underline{z})+(\underline{u}∗\underline{v})) <= (\underline{y}∗\underline{z}); PositiveToRight(Leq)⊳((\underline{x}∗\underline{z})+(\underline{u}∗\underline{v})) <= (\underline{y} ∗ \underline{z}) ≫ (\underline{x} ∗ \underline{z}) <= ((\underline{y} ∗ \underline{z}) + (−u(\underline{u} ∗ \underline{v})))⌉, p_{0}, c)]

[LessMultiplication(F)(Helper2) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: ∀\underline{v}: ˙¬ (0 <= \underline{u} ⇒ ˙¬ ( ˙¬ (0 = \underline{u})n)n)n ⊢ ˙¬ (0 <= \underline{v} ⇒ ˙¬ ( ˙¬ (0 = \underline{v})n)n)n ⊢ \underline{x} <= (\underline{y} + (−u\underline{u})) ⊢ 0 <= (\underline{z} + (−u\underline{v})) ⊢ (\underline{x} ∗ \underline{z}) <= ((\underline{y} ∗ \underline{z}) + (−u(\underline{u} ∗ \underline{v})))]

[LessMultiplication(F)(Helper2) →^{tex} “LessMultiplication(F)(Helper2)”]

[LessMultiplication(F)(Helper2) →^{pyk} “lemma lessMultiplication(F) helper2”]

LessMultiplication(F)(Helper)

[LessMultiplication(F)(Helper) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: ∀\underline{(ϵ)_{1}}: ∀\underline{(ϵ2)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)_{1}} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)_{1}})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)_{1}})))n)n ⊢ ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ2)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ2)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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}] + (−u\underline{(ϵ2)})))n)n ⊢ A4 @ \underline{m} ⊳ ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)_{1}} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)_{1}})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)_{1}})))n)n ≫ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)_{1}} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)_{1}})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)_{1}})))n)n; A4 @ \underline{m} ⊳ ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ2)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ2)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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}] + (−u\underline{(ϵ2)})))n)n ≫ ˙¬ ( ˙¬ (0 <= \underline{(ϵ2)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ2)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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}] + (−u\underline{(ϵ2)})))n)n; FirstConjunct ⊳ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)_{1}} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)_{1}})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)_{1}})))n)n ≫ ˙¬ (0 <= \underline{(ϵ)_{1}} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)_{1}})n)n)n; FirstConjunct ⊳ ˙¬ ( ˙¬ (0 <= \underline{(ϵ2)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ2)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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}] + (−u\underline{(ϵ2)})))n)n ≫ ˙¬ (0 <= \underline{(ϵ2)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ2)})n)n)n; PositiveFactors ⊳ ˙¬ (0 <= \underline{(ϵ)_{1}} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)_{1}})n)n)n ⊳ ˙¬ (0 <= \underline{(ϵ2)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ2)})n)n)n ≫ ˙¬ (0 <= (\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}) ⇒ ˙¬ ( ˙¬ (0 = (\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}))n)n)n; ∀\underline{m}: ∀\underline{n}: ∀\underline{(ϵ)_{1}}: ∀\underline{(ϵ2)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)_{1}} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)_{1}})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)_{1}})))n)n ⊢ ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ2)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ2)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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}] + (−u\underline{(ϵ2)})))n)n ⊢ \underline{n} <= \underline{m} ⊢ A4 @ \underline{m} ⊳ ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)_{1}} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)_{1}})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)_{1}})))n)n ≫ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)_{1}} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)_{1}})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)_{1}})))n)n; A4 @ \underline{m} ⊳ ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ2)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ2)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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}] + (−u\underline{(ϵ2)})))n)n ≫ ˙¬ ( ˙¬ (0 <= \underline{(ϵ2)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ2)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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}] + (−u\underline{(ϵ2)})))n)n; FirstConjunct ⊳ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)_{1}} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)_{1}})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)_{1}})))n)n ≫ ˙¬ (0 <= \underline{(ϵ)_{1}} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)_{1}})n)n)n; FirstConjunct ⊳ ˙¬ ( ˙¬ (0 <= \underline{(ϵ2)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ2)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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}] + (−u\underline{(ϵ2)})))n)n ≫ ˙¬ (0 <= \underline{(ϵ2)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ2)})n)n)n; SecondConjunct ⊳ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)_{1}} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)_{1}})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)_{1}})))n)n ≫ \underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)_{1}})); MP ⊳ \underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)_{1}})) ⊳ \underline{n} <= \underline{m} ≫ \underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)_{1}})); SecondConjunct ⊳ ˙¬ ( ˙¬ (0 <= \underline{(ϵ2)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ2)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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}] + (−u\underline{(ϵ2)})))n)n ≫ \underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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}] + (−u\underline{(ϵ2)})); MP ⊳ \underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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}] + (−u\underline{(ϵ2)})) ⊳ \underline{n} <= \underline{m} ≫ {ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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}] + (−u\underline{(ϵ2)})); NatType ≫ \underline{m} ∈ N; 0f ⊳ \underline{m} ∈ N ≫ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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}] = 0; subLeqLeft ⊳ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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}] = 0 ⊳ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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}] + (−u\underline{(ϵ2)})) ≫ 0 <= (\underline{(fz)}[\underline{m}] + (−u\underline{(ϵ2)})); LessMultiplication(F)(Helper2) ⊳ ˙¬ (0 <= \underline{(ϵ)_{1}} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)_{1}})n)n)n ⊳ ˙¬ (0 <= \underline{(ϵ2)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ2)})n)n)n ⊳ \underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)_{1}})) ⊳ 0 <= (\underline{(fz)}[\underline{m}] + (−u\underline{(ϵ2)})) ≫ (\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}]) <= ((\underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}]) + (−u(\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}))); TimesF(Sym) ≫ (\underline{(fx)}[\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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]; subLeqLeft ⊳ (\underline{(fx)}[\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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}] ⊳ (\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}]) <= ((\underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}]) + (−u(\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}))) ≫ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] <= ((\underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}]) + (−u(\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}))); TimesF(Sym) ≫ (\underline{(fy)}[\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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}]; eqAddition ⊳ (\underline{(fy)}[\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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}] ≫ ((\underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}]) + (−u(\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}))) = (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}] + (−u(\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}))); subLeqRight ⊳ ((\underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}]) + (−u(\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}))) = (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}] + (−u(\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}))) ⊳ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] <= ((\underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}]) + (−u(\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}))) ≫ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}] + (−u(\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}))); ∀\underline{m}: ∀\underline{n}: ∀\underline{(ϵ)_{1}}: ∀\underline{(ϵ2)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: Ded ⊳ ∀\underline{m}: ∀\underline{n}: ∀\underline{(ϵ)_{1}}: ∀\underline{(ϵ2)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)_{1}} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)_{1}})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)_{1}})))n)n ⊢ ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ2)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ2)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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}] + (−u\underline{(ϵ2)})))n)n ⊢ ˙¬ (0 <= (\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}) ⇒ ˙¬ ( ˙¬ (0 = (\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}))n)n)n ≫ ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)_{1}} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)_{1}})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)_{1}})))n)n ⇒ ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ2)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ2)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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}] + (−u\underline{(ϵ2)})))n)n ⇒ ˙¬ (0 <= (\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}) ⇒ ˙¬ ( ˙¬ (0 = (\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}))n)n)n; Ded ⊳ ∀\underline{m}: ∀\underline{n}: ∀\underline{(ϵ)_{1}}: ∀\underline{(ϵ2)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)_{1}} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)_{1}})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)_{1}})))n)n ⊢ ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ2)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ2)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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}] + (−u\underline{(ϵ2)})))n)n ⊢ \underline{n} <= \underline{m} ⊢ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}] + (−u(\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}))) ≫ ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)_{1}} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)_{1}})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)_{1}})))n)n ⇒ ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ2)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ2)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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}] + (−u\underline{(ϵ2)})))n)n ⇒ \underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}] + (−u(\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}))); JoinConjuncts(2conditions) ⊳ ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)_{1}} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)_{1}})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)_{1}})))n)n ⇒ ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ2)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ2)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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}] + (−u\underline{(ϵ2)})))n)n ⇒ ˙¬ (0 <= (\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}) ⇒ ˙¬ ( ˙¬ (0 = (\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}))n)n)n ⊳ ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)_{1}} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)_{1}})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)_{1}})))n)n ⇒ ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ2)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ2)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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}] + (−u\underline{(ϵ2)})))n)n ⇒ \underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}] + (−u(\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}))) ≫ ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)_{1}} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)_{1}})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)_{1}})))n)n ⇒ ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ2)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ2)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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}] + (−u\underline{(ϵ2)})))n)n ⇒ ˙¬ ( ˙¬ (0 <= (\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}) ⇒ ˙¬ ( ˙¬ (0 = (\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}))n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}] + (−u(\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}))))n)n⌉, p_{0}, c)]

[LessMultiplication(F)(Helper) →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: ∀\underline{(ϵ)_{1}}: ∀\underline{(ϵ2)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)_{1}} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)_{1}})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)_{1}})))n)n ⇒ ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ2)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ2)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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}] + (−u\underline{(ϵ2)})))n)n ⇒ ˙¬ ( ˙¬ (0 <= (\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}) ⇒ ˙¬ ( ˙¬ (0 = (\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}))n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}] + (−u(\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}))))n)n]

[LessMultiplication(F)(Helper) →^{tex} “LessMultiplication(F)(Helper)”] [LessMultiplication(F)(Helper) →^{pyk} “lemma lessMultiplication(F) helper”]

LessMultiplication(F)

[LessMultiplication(F) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: ∀\underline{(ϵ)_{1}}: ∀\underline{(ϵ2)}: ∀\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}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})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}] <= (\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)_{1}}: ˙¬ ( ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)_{1}} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)_{1}})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)_{1}})))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} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \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{(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}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fz)}[\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} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \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{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ (∀_{obj}\underline{(ϵ2)}: ˙¬ ( ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ2)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ2)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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}] + (−u\underline{(ϵ2)})))n)n)n)n)n)n; LessMultiplication(F)(Helper) ≫ ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)_{1}} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)_{1}})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)_{1}})))n)n ⇒ ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ2)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ2)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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}] + (−u\underline{(ϵ2)})))n)n ⇒ ˙¬ ( ˙¬ (0 <= (\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}) ⇒ ˙¬ ( ˙¬ (0 = (\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}))n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}] + (−u(\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}))))n)n; TwiceExistMP2 ⊳ ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)_{1}} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)_{1}})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)_{1}})))n)n ⇒ ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ2)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ2)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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}] + (−u\underline{(ϵ2)})))n)n ⇒ ˙¬ ( ˙¬ (0 <= (\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}) ⇒ ˙¬ ( ˙¬ (0 = (\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}))n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}] + (−u(\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}))))n)n ⊳ ˙¬ (∀_{obj}\underline{(ϵ)_{1}}: ˙¬ ( ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)_{1}} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)_{1}})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] <= (\underline{(fy)}[\underline{m}] + (−u\underline{(ϵ)_{1}})))n)n)n)n)n)n ⊳ ˙¬ (∀_{obj}\underline{(ϵ2)}: ˙¬ ( ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ2)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ2)})n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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}] + (−u\underline{(ϵ2)})))n)n)n)n)n)n ≫ ˙¬ ( ˙¬ (0 <= (\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}) ⇒ ˙¬ ( ˙¬ (0 = (\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}))n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}] + (−u(\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}))))n)n; Gen ⊳ ˙¬ ( ˙¬ (0 <= (\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}) ⇒ ˙¬ ( ˙¬ (0 = (\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}))n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}] + (−u(\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}))))n)n ≫ ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= (\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}) ⇒ ˙¬ ( ˙¬ (0 = (\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}))n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}] + (−u(\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}))))n)n; IntroExist @ \underline{n} ⊳ ∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= (\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}) ⇒ ˙¬ ( ˙¬ (0 = (\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}))n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}] + (−u(\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}))))n)n ≫ ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= (\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}) ⇒ ˙¬ ( ˙¬ (0 = (\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}))n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}] + (−u(\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}))))n)n)n)n; IntroExist @(\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}) ⊳ ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ (0 <= (\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}) ⇒ ˙¬ ( ˙¬ (0 = (\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}))n)n)n ⇒ ˙¬ (\underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}] + (−u(\underline{(ϵ)_{1}} ∗ \underline{(ϵ2)}))))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} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}] + (−u\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} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\underline{m}] + (−u\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} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\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} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n⌉, p_{0}, c)] [LessMultiplication(F) →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: ∀\underline{(ϵ)_{1}}: ∀\underline{(ϵ2)}: ∀\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}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})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}] <= (\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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n]

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

[LessMultiplication(F) →^{pyk} “lemma lessMultiplication(F)”]

LessMultiplication(R)

[LessMultiplication(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}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})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}] <= (\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} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \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{(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}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}] <= (\underline{(fz)}[\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; LessMultiplication(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{(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{(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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\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} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n; eqReflexivity ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}[\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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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\} ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n; eqReflexivity ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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\}; 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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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\} ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n⌉, p_{0}, c)]

[LessMultiplication(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}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})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}] <= (\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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n]

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

[LessMultiplication(R) →^{pyk} “lemma lessMultiplication(R)”]

LeqMultiplication(R)

[LeqMultiplication(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}\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\} ⊢==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} ⇒ ˙¬ (|(\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} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)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} ⇒ ˙¬ (|(\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}\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}] ∗ \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}: ˙¬ (e_{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}: ˙¬ (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) ≫ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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) ≫ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(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}: ˙¬ (e_{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}\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}: ˙¬ (e_{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}: ˙¬ (e_{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}\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}: ˙¬ (e_{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}: ˙¬ (e_{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\}; 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}\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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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\}; ==Transitivity ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] ∗ \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}: ˙¬ (e_{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\}; == Transitivity ⊳ {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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\}; ==Transitivity ⊳ \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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\}; 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} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(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}: ˙¬ (e_{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\} ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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\}; ∀\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\} ⊢ EqMultiplication(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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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\}; 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} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = {ph ∈ P({ph ∈ P({ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(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}: ˙¬ (e_{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\} ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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\}; ∀\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}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})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}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)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{(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{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))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{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)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{(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{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))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; LessMultiplication(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{(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{(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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\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}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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\}; ∀\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} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)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\} ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}\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\} ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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\}; 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\} ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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\}; 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} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \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{(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{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))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}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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\} ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \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{(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{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))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}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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\}; ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \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{(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} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)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\} ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(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} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \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{(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} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)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\} ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \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{(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} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)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\}; 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\}; ExpandDisjuncts ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \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{(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} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)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\} ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(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} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)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\})n ⇒ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(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{(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{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n)n)n; From3Disjuncts ⊳ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)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\})n ⇒ ˙¬ (\{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(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{(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{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)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{(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} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\} = \{ph ∈ P(\{ph ∈ P(\{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}) | ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ f_{Ph} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ f_{Ph} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ f_{Ph} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ f_{Ph})n)n)n)n\}) | ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|(\underline{(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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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\} ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \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{(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{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))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}\underline{m}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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\} ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}[\overline{m}] <= (\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}] ∗ \underline{(fz)}[\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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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\}⌉, p_{0}, c)]

[LeqMultiplication(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}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\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} ⇒ ˙¬ (|(\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)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\} ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fz)}[\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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fy)}[\underline{m}] ∗ \underline{(fz)}[\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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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}: ˙¬ (e_{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\}]

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

[LeqMultiplication(R) →^{pyk} “lemma leqMultiplication(R)”]

PlusAssociativity(F)

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

[PlusAssociativity(F) →^{pyk} “lemma plusAssociativity(F)”]

Plus0(F)

[Plus0(F) →^{tex} “Plus0(F)”]

[Plus0(F) →^{pyk} “lemma plus0(F)”]

PlusCommutativity(F)

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

[PlusCommutativity(F) →^{pyk} “lemma plusCommutativity(F)”]

TimesAssociativity(F)

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

[TimesAssociativity(F) →^{pyk} “lemma timesAssociativity(F)”]

Times1f

[Times1f →^{tex} “Times1f”]

[Times1f →^{pyk} “lemma times1f”]

Cauchy(2)(Helper)

[Cauchy(2)(Helper) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{(n1)}: ∀\underline{(n2)}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n ⊢ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n2)} <= \underline{(v1)} ⇒ \underline{(n2)} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n ⊢ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⊢ if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{(v1)} ⊢ if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{(v2)} ⊢ MaxLeq(1) ≫ \underline{(n1)} <= if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}); leqTransitivity ⊳ \underline{(n1)} <= if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) ⊳ if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{(v1)} ≫ \underline{(n1)} <= \underline{(v1)}; leqTransitivity ⊳ \underline{(n1)} <= if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) ⊳ if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{(v2)} ≫ \underline{(n1)} <= \underline{(v2)}; MaxLeq(2) ≫ \underline{(n2)} <= if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}); leqTransitivity ⊳ \underline{(n2)} <= if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) ⊳ if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{(v1)} ≫ \underline{(n2)} <= \underline{(v1)}; leqTransitivity ⊳ \underline{(n2)} <= if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) ⊳ if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{(v2)} ≫ \underline{(n2)} <= \underline{(v2)}; A4 @ \underline{(v1)} ⊳ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n ≫ ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n; A4 @ \underline{(v2)} ⊳ ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n ≫ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n; MP3 ⊳ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n ⊳ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⊳ \underline{(n1)} <= \underline{(v1)} ⊳ \underline{(n1)} <= \underline{(v2)} ≫ ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n; A4 @ \underline{(v1)} ⊳ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n2)} <= \underline{(v1)} ⇒ \underline{(n2)} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n ≫ ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n2)} <= \underline{(v1)} ⇒ \underline{(n2)} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n; A4 @ \underline{(v2)} ⊳ ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n2)} <= \underline{(v1)} ⇒ \underline{(n2)} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n ≫ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n2)} <= \underline{(v1)} ⇒ \underline{(n2)} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n; MP3 ⊳ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n2)} <= \underline{(v1)} ⇒ \underline{(n2)} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n ⊳ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n⊳\underline{(n2)} <= \underline{(v1)}⊳\underline{(n2)} <= \underline{(v2)} ≫ ˙¬ (|(\underline{(fy)}[\underline{(v1)}]+(−u\underline{(fy)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n; JoinConjuncts ⊳ ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n ⊳ ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n ≫ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n)n)n; ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{(n1)}: ∀\underline{(n2)}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: Ded ⊳ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{(n1)}: ∀\underline{(n2)}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n ⊢ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n2)} <= \underline{(v1)} ⇒ \underline{(n2)} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n ⊢ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⊢ if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{(v1)} ⊢ if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{(v2)} ⊢ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n)n)n ≫ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n ⇒ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n2)} <= \underline{(v1)} ⇒ \underline{(n2)} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n ⇒ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{(v1)} ⇒ if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n)n)n⌉, p_{0}, c)]

[Cauchy(2)(Helper) →^{stmt} SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{(n1)}: ∀\underline{(n2)}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n ⇒ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n2)} <= \underline{(v1)} ⇒ \underline{(n2)} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n ⇒ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{(v1)} ⇒ if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n)n)n]

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

[Cauchy(2)(Helper) →^{pyk} “lemma 2cauchy helper”]

Cauchy(2)

[Cauchy(2) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{n}: ∀\underline{(n1)}: ∀\underline{(n2)}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: Cauchy ≫ ∀_{obj}\underline{(ϵ)}: ˙¬ (∀_{obj}\underline{(n1)}: ˙¬ (∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n)n)n; A4 @ \underline{(ϵ)} ⊳ ∀_{obj}\underline{(ϵ)}: ˙¬ (∀_{obj}\underline{(n1)}: ˙¬ (∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n)n)n ≫ ˙¬ (∀_{obj}\underline{(n1)}: ˙¬ (∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n)n)n; Cauchy ≫ ∀_{obj}\underline{(ϵ)}: ˙¬ (∀_{obj}\underline{(n2)}: ˙¬ (∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n2)} <= \underline{(v1)} ⇒ \underline{(n2)} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n)n)n; A4 @ \underline{(ϵ)} ⊳ ∀_{obj}\underline{(ϵ)}: ˙¬ (∀_{obj}\underline{(n2)}: ˙¬ (∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n2)} <= \underline{(v1)} ⇒ \underline{(n2)} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n)n)n ≫ ˙¬ (∀_{obj}\underline{(n2)}: ˙¬ (∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n2)} <= \underline{(v1)} ⇒ \underline{(n2)} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n)n)n; Cauchy(2)(Helper) ≫ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n ⇒ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n2)} <= \underline{(v1)} ⇒ \underline{(n2)} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n ⇒ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{(v1)} ⇒ if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n)n)n; ExistMP2 ⊳ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n ⇒ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n2)} <= \underline{(v1)} ⇒ \underline{(n2)} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n ⇒ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{(v1)} ⇒ if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n)n)n ⊳ ˙¬ (∀_{obj}\underline{(n1)}: ˙¬ (∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n)n)n ⊳ ˙¬ (∀_{obj}\underline{(n2)}: ˙¬ (∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n2)} <= \underline{(v1)} ⇒ \underline{(n2)} <= \underline{(v2)} ⇒ ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n)n)n ≫ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{(v1)} ⇒ if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n)n)n; Gen ⊳ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{(v1)} ⇒ if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n)n)n ≫ ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{(v1)} ⇒ if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n)n)n; Gen ⊳ ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{(v1)} ⇒ if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n)n)n ≫ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{(v1)} ⇒ if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n)n)n; IntroExist @ if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) ⊳ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{(v1)} ⇒ if(\underline{(n2)} <= \underline{(n1)}, \underline{(n1)}, \underline{(n2)}) <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n)n)n ≫ ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{n} <= \underline{(v1)} ⇒ \underline{n} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n)n)n)n)n; Gen⊳ ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{n} <= \underline{(v1)} ⇒ \underline{n} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n)n)n)n)n ≫ ∀_{obj}\underline{(ϵ)}: ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{n} <= \underline{(v1)} ⇒ \underline{n} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n)n)n)n)n⌉, p_{0}, c)]

[Cauchy(2) →^{stmt} SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{n}: ∀\underline{(n1)}: ∀\underline{(n2)}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀_{obj}\underline{(ϵ)}: ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{n} <= \underline{(v1)} ⇒ \underline{n} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n)n)n)n)n]

[Cauchy(2) →^{tex} “Cauchy(2)”]

[Cauchy(2) →^{pyk} “lemma 2cauchy”]

ReciprocalFnonzero

[ReciprocalFnonzero →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{(m1)}: ∀\underline{(fx)}: ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⊢ NatType ≫ \underline{m} ∈ N; ReciprocalIsRationalSeries ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ {ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\})n)n)n)n; MemberOfSeries ⊳ \underline{m} ∈ N ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\})n)n)n)n ≫ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}]\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}; Repetition ⊳ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}]\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\} ≫ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}]\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}; Separation2formula(2) ⊳ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}]\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\} ≫ ˙¬ (∀_{obj}\underline{(m1)}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{(m1)}] = 0)n ⇒ ˙¬ (\{\{\underline{m}, \underline{m}\}, \{\underline{m}, \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}]\}\} = \{\{\underline{(m1)}, \underline{(m1)}\}, \{\underline{(m1)}, rec\underline{(fx)}[\underline{(m1)}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{(m1)}] = 0 ⇒ ˙¬ (\{\{\underline{m}, \underline{m}\}, \{\underline{m}, \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}]\}\} = \{\{\underline{(m1)}, \underline{(m1)}\}, \{\underline{(m1)}, 0\}\})n)n)n)n; ∀\underline{m}: ∀\underline{(m1)}: ∀\underline{(fx)}: ˙¬ (\underline{(fx)}[\underline{(m1)}] = 0)n ⊢ ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{(m1)}] = 0)n ⇒ ˙¬ (\{\{\underline{m}, \underline{m}\}, \{\underline{m}, \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}]\}\} = \{\{\underline{(m1)}, \underline{(m1)}\}, \{\underline{(m1)}, rec\underline{(fx)}[\underline{(m1)}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{(m1)}] = 0 ⇒ ˙¬ (\{\{\underline{m}, \underline{m}\}, \{\underline{m}, \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}]\}\} = \{\{\underline{(m1)}, \underline{(m1)}\}, \{\underline{(m1)}, 0\}\})n)n ⊢ ToNegatedAnd(1) ⊳ ˙¬ (\underline{(fx)}[\underline{(m1)}] = 0)n ≫ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{(m1)}] = 0 ⇒ ˙¬ (\{\{\underline{m}, \underline{m}\}, \{\underline{m}, \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}]\}\} = \{\{\underline{(m1)}, \underline{(m1)}\}, \{\underline{(m1)}, 0\}\})n)n)n; NegateDisjunct2 ⊳ ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{(m1)}] = 0)n ⇒ ˙¬ (\{\{\underline{m}, \underline{m}\}, \{\underline{m}, \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}]\}\} = \{\{\underline{(m1)}, \underline{(m1)}\}, \{\underline{(m1)}, rec\underline{(fx)}[\underline{(m1)}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{(m1)}] = 0 ⇒ ˙¬ (\{\{\underline{m}, \underline{m}\}, \{\underline{m}, \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}]\}\} = \{\{\underline{(m1)}, \underline{(m1)}\}, \{\underline{(m1)}, 0\}\})n)n ⊳ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{(m1)}] = 0 ⇒ ˙¬ (\{\{\underline{m}, \underline{m}\}, \{\underline{m}, \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}]\}\} = \{\{\underline{(m1)}, \underline{(m1)}\}, \{\underline{(m1)}, 0\}\})n)n)n ≫ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{(m1)}] = 0)n ⇒ ˙¬ (\{\{\underline{m}, \underline{m}\}, \{\underline{m}, \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}]\}\} = \{\{\underline{(m1)}, \underline{(m1)}\}, \{\underline{(m1)}, rec\underline{(fx)}[\underline{(m1)}]\}\})n)n; SecondConjunct ⊳ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{(m1)}] = 0)n ⇒ ˙¬ (\{\{\underline{m}, \underline{m}\}, \{\underline{m}, \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}]\}\} = \{\{\underline{(m1)}, \underline{(m1)}\}, \{\underline{(m1)}, rec\underline{(fx)}[\underline{(m1)}]\}\})n)n ≫ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}]\}\} = \{\{\underline{(m1)}, \underline{(m1)}\}, \{\underline{(m1)}, rec\underline{(fx)}[\underline{(m1)}]\}\}; FromOrderedPair(2) ⊳ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}]\}\} = \{\{\underline{(m1)}, \underline{(m1)}\}, \{\underline{(m1)}, rec\underline{(fx)}[\underline{(m1)}]\}\} ≫ {ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}] = rec\underline{(fx)}[\underline{(m1)}]; FromOrderedPair(1) ⊳ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}]\}\} = \{\{\underline{(m1)}, \underline{(m1)}\}, \{\underline{(m1)}, rec\underline{(fx)}[\underline{(m1)}]\}\} ≫ \underline{m} = \underline{(m1)}; SameSeries ⊳ \underline{m} = \underline{(m1)} ≫ \underline{(fx)}[\underline{m}] = \underline{(fx)}[\underline{(m1)}]; eqSymmetry ⊳ \underline{(fx)}[\underline{m}] = \underline{(fx)}[\underline{(m1)}] ≫ \underline{(fx)}[\underline{(m1)}] = \underline{(fx)}[\underline{m}]; SameReciprocal ⊳ ˙¬ (\underline{(fx)}[\underline{(m1)}] = 0)n ⊳ \underline{(fx)}[\underline{(m1)}] = \underline{(fx)}[\underline{m}] ≫ rec\underline{(fx)}[\underline{(m1)}] = rec\underline{(fx)}[\underline{m}]; eqTransitivity ⊳ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}] = rec\underline{(fx)}[\underline{(m1)}] ⊳ rec\underline{(fx)}[\underline{(m1)}] = rec\underline{(fx)}[\underline{m}] ≫ {ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}] = rec\underline{(fx)}[\underline{m}]; Ded ⊳ ∀\underline{m}: ∀\underline{(m1)}: ∀\underline{(fx)}: ˙¬ (\underline{(fx)}[\underline{(m1)}] = 0)n ⊢ ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{(m1)}] = 0)n ⇒ ˙¬ (\{\{\underline{m}, \underline{m}\}, \{\underline{m}, \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}]\}\} = \{\{\underline{(m1)}, \underline{(m1)}\}, \{\underline{(m1)}, rec\underline{(fx)}[\underline{(m1)}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{(m1)}] = 0 ⇒ ˙¬ (\{\{\underline{m}, \underline{m}\}, \{\underline{m}, \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}]\}\} = \{\{\underline{(m1)}, \underline{(m1)}\}, \{\underline{(m1)}, 0\}\})n)n ⊢ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}] = rec\underline{(fx)}[\underline{m}] ≫ ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{(m1)}] = 0)n ⇒ ˙¬ (\{\{\underline{m}, \underline{m}\}, \{\underline{m}, \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}]\}\} = \{\{\underline{(m1)}, \underline{(m1)}\}, \{\underline{(m1)}, rec\underline{(fx)}[\underline{(m1)}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{(m1)}] = 0 ⇒ ˙¬ (\{\{\underline{m}, \underline{m}\}, \{\underline{m}, \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}]\}\} = \{\{\underline{(m1)}, \underline{(m1)}\}, \{\underline{(m1)}, 0\}\})n)n ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}] = rec\underline{(fx)}[\underline{m}]; MP ⊳ ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{(m1)}] = 0)n ⇒ ˙¬ (\{\{\underline{m}, \underline{m}\}, \{\underline{m}, \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}]\}\} = \{\{\underline{(m1)}, \underline{(m1)}\}, \{\underline{(m1)}, rec\underline{(fx)}[\underline{(m1)}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{(m1)}] = 0 ⇒ ˙¬ (\{\{\underline{m}, \underline{m}\}, \{\underline{m}, \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}]\}\} = \{\{\underline{(m1)}, \underline{(m1)}\}, \{\underline{(m1)}, 0\}\})n)n ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}] = rec\underline{(fx)}[\underline{m}] ⊳ ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ≫ ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{(m1)}] = 0)n ⇒ ˙¬ (\{\{\underline{m}, \underline{m}\}, \{\underline{m}, \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}]\}\} = \{\{\underline{(m1)}, \underline{(m1)}\}, \{\underline{(m1)}, rec\underline{(fx)}[\underline{(m1)}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{(m1)}] = 0 ⇒ ˙¬ (\{\{\underline{m}, \underline{m}\}, \{\underline{m}, \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}]\}\} = \{\{\underline{(m1)}, \underline{(m1)}\}, \{\underline{(m1)}, 0\}\})n)n ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}] = rec\underline{(fx)}[\underline{m}]; ExistMP ⊳ ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{(m1)}] = 0)n ⇒ ˙¬ (\{\{\underline{m}, \underline{m}\}, \{\underline{m}, \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}]\}\} = \{\{\underline{(m1)}, \underline{(m1)}\}, \{\underline{(m1)}, rec\underline{(fx)}[\underline{(m1)}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{(m1)}] = 0 ⇒ ˙¬ (\{\{\underline{m}, \underline{m}\}, \{\underline{m}, \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}]\}\} = \{\{\underline{(m1)}, \underline{(m1)}\}, \{\underline{(m1)}, 0\}\})n)n ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}] = rec\underline{(fx)}[\underline{m}] ⊳ ˙¬ (∀_{obj}\underline{(m1)}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{(m1)}] = 0)n ⇒ ˙¬ (\{\{\underline{m}, \underline{m}\}, \{\underline{m}, \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}]\}\} = \{\{\underline{(m1)}, \underline{(m1)}\}, \{\underline{(m1)}, rec\underline{(fx)}[\underline{(m1)}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{(m1)}] = 0 ⇒ ˙¬ (\{\{\underline{m}, \underline{m}\}, \{\underline{m}, \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}]\}\} = \{\{\underline{(m1)}, \underline{(m1)}\}, \{\underline{(m1)}, 0\}\})n)n)n)n ≫ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}] = rec\underline{(fx)}[\underline{m}]⌉, p_{0}, c)]

[ReciprocalFnonzero →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{(m1)}: ∀\underline{(fx)}: ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⊢ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}] = rec\underline{(fx)}[\underline{m}]]

[ReciprocalFnonzero →^{tex} “ReciprocalFnonzero”]

[ReciprocalFnonzero →^{pyk} “lemma reciprocalF nonzero”]

(Eventually = f)2sameF(Helper)

[(Eventually = f)2sameF(Helper) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀_{obj}\underline{m}: \underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}] ⊢ A4 @ \underline{m} ⊳ ∀_{obj}\underline{m}: \underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}] ≫ \underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}]; ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⊢ \underline{n} <= \underline{m} ⊢ MP ⊳ \underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}] ⊳ \underline{n} <= \underline{m} ≫ \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}]; PositiveToLeft(Eq)(1term) ⊳ \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}] ≫ (\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}])) = 0; SameNumerical ⊳ (\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}])) = 0 ≫ |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| = |0|; |0| = 0 ≫ |0| = 0; eqTransitivity ⊳ |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| = |0| ⊳ |0| = 0 ≫ |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| = 0; eqSymmetry ⊳ |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| = 0 ≫ 0 = |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))|; SubLessLeft ⊳ 0 = |(\underline{(fx)}[\underline{m}] + (−u\underline{(fy)}[\underline{m}]))| ⊳ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ≫ ˙¬ (|(\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; ∀\underline{m}: ∀\underline{n}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: Ded ⊳ ∀\underline{m}: ∀\underline{n}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀_{obj}\underline{m}: \underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] = \underline{(fy)}[\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 ≫ ∀_{obj}\underline{m}: \underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] = \underline{(fy)}[\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⌉, p_{0}, c)]

[(Eventually = f)2sameF(Helper) →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀_{obj}\underline{m}: \underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] = \underline{(fy)}[\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]

[(Eventually = f)2sameF(Helper) →^{tex} “(Eventually=f)2sameF(Helper)”]

[(Eventually = f)2sameF(Helper) →^{pyk} “lemma eventually=f to sameF helper”]

(Eventually = f)2sameF

[(Eventually = f)2sameF →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{m}: \underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}])n)n ⊢ (Eventually = f)2sameF(Helper) ≫ ∀_{obj}\underline{m}: \underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] = \underline{(fy)}[\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; ExistMP ⊳ ∀_{obj}\underline{m}: \underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] = \underline{(fy)}[\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 ⊳ ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{m}: \underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}])n)n ≫ ˙¬ (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; Gen ⊳ ˙¬ (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 ≫ ∀_{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; IntroExist @ \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 ≫ ˙¬ (∀_{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; Gen ⊳ ˙¬ (∀_{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 ≫ ∀_{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; 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 ≫ ∀_{obj}\overline{(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}] + (−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⌉, p_{0}, c)]

[(Eventually = f)2sameF →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{m}: \underline{n} <= \underline{m} ⇒ \underline{(fx)}[\underline{m}] = \underline{(fy)}[\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} ⇒ ˙¬ (|(\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]

[(Eventually = f)2sameF →^{tex} “(Eventually=f)2sameF”]

[(Eventually = f)2sameF →^{pyk} “lemma eventually=f to sameF”]

FromNotSameF(Strong)(Helper2)

[FromNotSameF(Strong)(Helper2) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: ∀\underline{v}: \underline{v} <= |(\underline{x} + (−u\underline{z}))| ⊢ ˙¬ (|(\underline{x} + (−u\underline{y}))| <= (rec(1 + 1) ∗ \underline{v}) ⇒ ˙¬ ( ˙¬ (|(\underline{x} + (−u\underline{y}))| = (rec(1 + 1) ∗ \underline{v}))n)n)n ⊢ ˙¬ (|(\underline{z} + (−u\underline{u}))| <= (rec(1 + 1) ∗ \underline{v}) ⇒ ˙¬ ( ˙¬ (|(\underline{z} + (−u\underline{u}))| = (rec(1 + 1) ∗ \underline{v}))n)n)n ⊢ LessNegated ⊳ ˙¬ (|(\underline{x} + (−u\underline{y}))| <= (rec(1 + 1) ∗ \underline{v}) ⇒ ˙¬ ( ˙¬ (|(\underline{x} + (−u\underline{y}))| = (rec(1 + 1) ∗ \underline{v}))n)n)n ≫ ˙¬ ((−u(rec(1 + 1) ∗ \underline{v})) <= (−u|(\underline{x} + (−u\underline{y}))|) ⇒ ˙¬ ( ˙¬ ((−u(rec(1 + 1) ∗ \underline{v})) = (−u|(\underline{x} + (−u\underline{y}))|))n)n)n; NumericalDifference ≫ |(\underline{z} + (−u\underline{u}))| = |(\underline{u} + (−u\underline{z}))|; SubLessLeft ⊳ |(\underline{z} + (−u\underline{u}))| = |(\underline{u} + (−u\underline{z}))| ⊳ ˙¬ (|(\underline{z} + (−u\underline{u}))| <= (rec(1 + 1) ∗ \underline{v}) ⇒ ˙¬ ( ˙¬ (|(\underline{z} + (−u\underline{u}))| = (rec(1 + 1) ∗ \underline{v}))n)n)n ≫ ˙¬ (|(\underline{u} + (−u\underline{z}))| <= (rec(1 + 1) ∗ \underline{v}) ⇒ ˙¬ ( ˙¬ (|(\underline{u} + (−u\underline{z}))| = (rec(1 + 1) ∗ \underline{v}))n)n)n; LessNegated ⊳ ˙¬ (|(\underline{u} + (−u\underline{z}))| <= (rec(1 + 1) ∗ \underline{v}) ⇒ ˙¬ ( ˙¬ (|(\underline{u} + (−u\underline{z}))| = (rec(1 + 1) ∗ \underline{v}))n)n)n ≫ ˙¬ ((−u(rec(1 + 1) ∗ \underline{v})) <= (−u|(\underline{u} + (−u\underline{z}))|) ⇒ ˙¬ ( ˙¬ ((−u(rec(1 + 1) ∗ \underline{v})) = (−u|(\underline{u} + (−u\underline{z}))|))n)n)n; AddEquations(Less) ⊳ ˙¬ ((−u(rec(1 + 1) ∗ \underline{v})) <= (−u|(\underline{x} + (−u\underline{y}))|) ⇒ ˙¬ ( ˙¬ ((−u(rec(1 + 1) ∗ \underline{v})) = (−u|(\underline{x} + (−u\underline{y}))|))n)n)n ⊳ ˙¬ ((−u(rec(1 + 1) ∗ \underline{v})) <= (−u|(\underline{u} + (−u\underline{z}))|) ⇒ ˙¬ ( ˙¬ ((−u(rec(1 + 1) ∗ \underline{v})) = (−u|(\underline{u} + (−u\underline{z}))|))n)n)n ≫ ˙¬ (((−u(rec(1 + 1) ∗ \underline{v})) + (−u(rec(1 + 1) ∗ \underline{v}))) <= ((−u|(\underline{x}+(−u\underline{y}))|)+(−u|(\underline{u}+(−u\underline{z}))|)) ⇒ ˙¬ ( ˙¬ (((−u(rec(1+1)∗\underline{v}))+(−u(rec(1+ 1) ∗ \underline{v}))) = ((−u|(\underline{x} + (−u\underline{y}))|) + (−u|(\underline{u} + (−u\underline{z}))|)))n)n)n; TwoHalves ≫ ((rec(1+1)∗\underline{v})+(rec(1+1)∗\underline{v})) = \underline{v}; EqNegated⊳((rec(1+1)∗\underline{v})+(rec(1+1)∗\underline{v})) = \underline{v} ≫ (−u((rec(1 + 1) ∗ \underline{v}) + (rec(1 + 1) ∗ \underline{v}))) = (−u\underline{v}); −x − y = −(x + y) ≫ ((−u(rec(1 + 1) ∗ \underline{v})) + (−u(rec(1 + 1) ∗ \underline{v}))) = (−u((rec(1 + 1) ∗ \underline{v}) + (rec(1 + 1) ∗ \underline{v}))); eqTransitivity ⊳ ((−u(rec(1 + 1) ∗ \underline{v})) + (−u(rec(1 + 1) ∗ \underline{v}))) = (−u((rec(1 + 1) ∗ \underline{v}) + (rec(1 + 1) ∗ \underline{v}))) ⊳ (−u((rec(1 + 1) ∗ \underline{v}) + (rec(1 + 1) ∗ \underline{v}))) = (−u\underline{v}) ≫ ((−u(rec(1 + 1) ∗ \underline{v})) + (−u(rec(1 + 1) ∗ \underline{v}))) = (−u\underline{v}); SubLessLeft ⊳ ((−u(rec(1 + 1) ∗ \underline{v})) + (−u(rec(1 + 1) ∗ \underline{v}))) = (−u\underline{v}) ⊳ ˙¬ (((−u(rec(1 + 1) ∗ \underline{v})) + (−u(rec(1 + 1) ∗ \underline{v}))) <= ((−u|(\underline{x} + (−u\underline{y}))|) + (−u|(\underline{u} + (−u\underline{z}))|)) ⇒ ˙¬ ( ˙¬ (((−u(rec(1 + 1) ∗ \underline{v})) + (−u(rec(1 + 1) ∗ \underline{v}))) = ((−u|(\underline{x} + (−u\underline{y}))|) + (−u|(\underline{u} + (−u\underline{z}))|)))n)n)n ≫ ˙¬ ((−u\underline{v}) <= ((−u|(\underline{x} + (−u\underline{y}))|) + (−u|(\underline{u} + (−u\underline{z}))|)) ⇒ ˙¬ ( ˙¬ ((−u\underline{v}) = ((−u|(\underline{x} + (−u\underline{y}))|) + (−u|(\underline{u} + (−u\underline{z}))|)))n)n)n; plusCommutativity ≫ ((−u|(\underline{x} + (−u\underline{y}))|) + (−u|(\underline{u} + (−u\underline{z}))|)) = ((−u|(\underline{u} + (−u\underline{z}))|) + (−u|(\underline{x} + (−u\underline{y}))|)); SubLessRight ⊳ ((−u|(\underline{x} + (−u\underline{y}))|) + (−u|(\underline{u} + (−u\underline{z}))|)) = ((−u|(\underline{u} + (−u\underline{z}))|) + (−u|(\underline{x} + (−u\underline{y}))|)) ⊳ ˙¬ ((−u\underline{v}) <= ((−u|(\underline{x} + (−u\underline{y}))|) + (−u|(\underline{u} + (−u\underline{z}))|)) ⇒ ˙¬ ( ˙¬ ((−u\underline{v}) = ((−u|(\underline{x} + (−u\underline{y}))|) + (−u|(\underline{u} + (−u\underline{z}))|)))n)n)n ≫ ˙¬ ((−u\underline{v}) <= ((−u|(\underline{u} + (−u\underline{z}))|) + (−u|(\underline{x} + (−u\underline{y}))|)) ⇒ ˙¬ ( ˙¬ ((−u\underline{v}) = ((−u|(\underline{u} + (−u\underline{z}))|) + (−u|(\underline{x} + (−u\underline{y}))|)))n)n)n; AddEquations(LeqLess) ⊳ \underline{v} <= |(\underline{x} + (−u\underline{z}))| ⊳ ˙¬ ((−u\underline{v}) <= ((−u|(\underline{u} + (−u\underline{z}))|) + (−u|(\underline{x} + (−u\underline{y}))|)) ⇒ ˙¬ ( ˙¬ ((−u\underline{v}) = ((−u|(\underline{u} + (−u\underline{z}))|) + (−u|(\underline{x} + (−u\underline{y}))|)))n)n)n ≫ ˙¬ ((\underline{v} + (−u\underline{v})) <= (|(\underline{x} + (−u\underline{z}))| + ((−u|(\underline{u} + (−u\underline{z}))|) + (−u|(\underline{x} + (−u\underline{y}))|))) ⇒ ˙¬ ( ˙¬ ((\underline{v} + (−u\underline{v})) = (|(\underline{x} + (−u\underline{z}))| + ((−u|(\underline{u} + (−u\underline{z}))|) + (−u|(\underline{x} + (−u\underline{y}))|))))n)n)n; Negative ≫ (\underline{v} + (−u\underline{v})) = 0; SubLessLeft ⊳ (\underline{v} + (−u\underline{v})) = 0 ⊳ ˙¬ ((\underline{v} + (−u\underline{v})) <= (|(\underline{x} + (−u\underline{z}))| + ((−u|(\underline{u} + (−u\underline{z}))|) + (−u|(\underline{x} + (−u\underline{y}))|))) ⇒ ˙¬ ( ˙¬ ((\underline{v} + (−u\underline{v})) = (|(\underline{x} + (−u\underline{z}))| + ((−u|(\underline{u} + (−u\underline{z}))|) + (−u|(\underline{x} + (−u\underline{y}))|))))n)n)n ≫ ˙¬ (0 <= (|(\underline{x} + (−u\underline{z}))| + ((−u|(\underline{u} + (−u\underline{z}))|) + (−u|(\underline{x} + (−u\underline{y}))|))) ⇒ ˙¬ ( ˙¬ (0 = (|(\underline{x} + (−u\underline{z}))| + ((−u|(\underline{u} + (−u\underline{z}))|) + (−u|(\underline{x} + (−u\underline{y}))|))))n)n)n; plusAssociativity ≫ ((|(\underline{x} + (−u\underline{z}))| + (−u|(\underline{u} + (−u\underline{z}))|)) + (−u|(\underline{x} + (−u\underline{y}))|)) = (|(\underline{x} + (−u\underline{z}))| + ((−u|(\underline{u} + (−u\underline{z}))|) + (−u|(\underline{x} + (−u\underline{y}))|))); eqSymmetry ⊳ ((|(\underline{x} + (−u\underline{z}))| + (−u|(\underline{u} + (−u\underline{z}))|)) + (−u|(\underline{x} + (−u\underline{y}))|)) = (|(\underline{x} + (−u\underline{z}))| + ((−u|(\underline{u} + (−u\underline{z}))|) + (−u|(\underline{x} + (−u\underline{y}))|))) ≫ (|(\underline{x} + (−u\underline{z}))| + ((−u|(\underline{u} + (−u\underline{z}))|) + (−u|(\underline{x} + (−u\underline{y}))|))) = ((|(\underline{x} + (−u\underline{z}))| + (−u|(\underline{u} + (−u\underline{z}))|)) + (−u|(\underline{x} + (−u\underline{y}))|)); SubLessRight ⊳ (|(\underline{x} + (−u\underline{z}))| + ((−u|(\underline{u} + (−u\underline{z}))|) + (−u|(\underline{x} + (−u\underline{y}))|))) = ((|(\underline{x} + (−u\underline{z}))| + (−u|(\underline{u} + (−u\underline{z}))|)) + (−u|(\underline{x} + (−u\underline{y}))|)) ⊳ ˙¬ (0 <= (|(\underline{x} + (−u\underline{z}))| + ((−u|(\underline{u} + (−u\underline{z}))|) + (−u|(\underline{x} + (−u\underline{y}))|))) ⇒ ˙¬ ( ˙¬ (0 = (|(\underline{x} + (−u\underline{z}))| + ((−u|(\underline{u} + (−u\underline{z}))|) + (−u|(\underline{x} + (−u\underline{y}))|))))n)n)n ≫ ˙¬ (0 <= ((|(\underline{x} + (−u\underline{z}))| + (−u|(\underline{u} + (−u\underline{z}))|)) + (−u|(\underline{x} + (−u\underline{y}))|)) ⇒ ˙¬ ( ˙¬ (0 = ((|(\underline{x} + (−u\underline{z}))| + (−u|(\underline{u} + (−u\underline{z}))|)) + (−u|(\underline{x} + (−u\underline{y}))|)))n)n)n; insertTwoMiddleTerms(Numerical) ≫ |(\underline{x} + (−u\underline{z}))| <= ((|(\underline{x} + (−u\underline{y}))| + |(\underline{y} + (−u\underline{u}))|) + |(\underline{u} + (−u\underline{z}))|); PositiveToLeft(Leq) ⊳ |(\underline{x} + (−u\underline{z}))| <= ((|(\underline{x} + (−u\underline{y}))| + |(\underline{y} + (−u\underline{u}))|) + |(\underline{u} + (−u\underline{z}))|) ≫ (|(\underline{x} + (−u\underline{z}))| + (−u|(\underline{u} + (−u\underline{z}))|)) <= (|(\underline{x} + (−u\underline{y}))| + |(\underline{y} + (−u\underline{u}))|); plusCommutativity ≫ (|(\underline{x} + (−u\underline{y}))| + |(\underline{y} + (−u\underline{u}))|) = (|(\underline{y} + (−u\underline{u}))| + |(\underline{x} + (−u\underline{y}))|); subLeqRight ⊳ (|(\underline{x} + (−u\underline{y}))| + |(\underline{y} + (−u\underline{u}))|) = (|(\underline{y} + (−u\underline{u}))| + |(\underline{x} + (−u\underline{y}))|) ⊳ (|(\underline{x} + (−u\underline{z}))| + (−u|(\underline{u} + (−u\underline{z}))|)) <= (|(\underline{x} + (−u\underline{y}))| + |(\underline{y} + (−u\underline{u}))|) ≫ (|(\underline{x} + (−u\underline{z}))| + (−u|(\underline{u} + (−u\underline{z}))|)) <= (|(\underline{y} + (−u\underline{u}))| + |(\underline{x} + (−u\underline{y}))|); PositiveToLeft(Leq) ⊳ (|(\underline{x} + (−u\underline{z}))| + (−u|(\underline{u} + (−u\underline{z}))|)) <= (|(\underline{y} + (−u\underline{u}))| + |(\underline{x} + (−u\underline{y}))|) ≫ ((|(\underline{x} + (−u\underline{z}))| + (−u|(\underline{u} + (−u\underline{z}))|)) + (−u|(\underline{x} + (−u\underline{y}))|)) <= |(\underline{y} + (−u\underline{u}))|; LessLeqTransitivity ⊳ ˙¬ (0 <= ((|(\underline{x} + (−u\underline{z}))| + (−u|(\underline{u} + (−u\underline{z}))|)) + (−u|(\underline{x} + (−u\underline{y}))|)) ⇒ ˙¬ ( ˙¬ (0 = ((|(\underline{x} + (−u\underline{z}))| + (−u|(\underline{u} + (−u\underline{z}))|)) + (−u|(\underline{x} + (−u\underline{y}))|)))n)n)n ⊳ ((|(\underline{x} + (−u\underline{z}))| + (−u|(\underline{u} + (−u\underline{z}))|)) + (−u|(\underline{x} + (−u\underline{y}))|)) <= |(\underline{y} + (−u\underline{u}))| ≫ ˙¬ (0 <= |(\underline{y}+(−u\underline{u}))| ⇒ ˙¬ ( ˙¬ (0 = |(\underline{y}+(−u\underline{u}))|)n)n)n; FromPositiveNumerical⊳ ˙¬ (0 <= |(\underline{y} + (−u\underline{u}))| ⇒ ˙¬ ( ˙¬ (0 = |(\underline{y} + (−u\underline{u}))|)n)n)n ≫ ˙¬ ((\underline{y} + (−u\underline{u})) = 0)n; NegativeToRight(Neq)(1term)⊳ ˙¬ ((\underline{y} +(−u\underline{u})) = 0)n ≫ ˙¬ (\underline{y} = \underline{u})n⌉, p_{0}, c)]

[FromNotSameF(Strong)(Helper2) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ∀\underline{u}: ∀\underline{v}: \underline{v} <= |(\underline{x} + (−u\underline{z}))| ⊢ ˙¬ (|(\underline{x} + (−u\underline{y}))| <= (rec(1 + 1) ∗ \underline{v}) ⇒ ˙¬ ( ˙¬ (|(\underline{x} + (−u\underline{y}))| = (rec(1 + 1) ∗ \underline{v}))n)n)n ⊢ ˙¬ (|(\underline{z} + (−u\underline{u}))| <= (rec(1 + 1) ∗ \underline{v}) ⇒ ˙¬ ( ˙¬ (|(\underline{z} + (−u\underline{u}))| = (rec(1 + 1) ∗ \underline{v}))n)n)n ⊢ ˙¬ (\underline{y} = \underline{u})n]

[FromNotSameF(Strong)(Helper2) →^{tex} “FromNotSameF(Strong)(Helper2)”]

[FromNotSameF(Strong)(Helper2) →^{pyk} “lemma fromNotSameF(Strong) helper2”]

FromNotSameF(Strong)(Helper)

[FromNotSameF(Strong)(Helper) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{m}: ∀\underline{(n1)}: ∀\underline{(n2)}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| = \underline{(ϵ)})n)n)n)n ⊢ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n)n)n ⊢ \underline{(n2)} <= \underline{m} ⊢ FromNegated(2 ∗ Imply) ⊳ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| = \underline{(ϵ)})n)n)n)n ≫ ˙¬ ( ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{(n2)})n)n ⇒ ˙¬ ( ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| = \underline{(ϵ)})n)n)n)n)n)n; FirstConjunct ⊳ ˙¬ ( ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{(n2)})n)n ⇒ ˙¬ ( ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| = \underline{(ϵ)})n)n)n)n)n)n ≫ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{(n2)})n)n; FirstConjunct ⊳ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{(n2)})n)n ≫ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n; SecondConjunct ⊳ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{(n2)})n)n ≫ \underline{(n1)} <= \underline{(n2)}; SecondConjunct ⊳ ˙¬ ( ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ ˙¬ (\underline{(n1)} <= \underline{(n2)})n)n ⇒ ˙¬ ( ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| = \underline{(ϵ)})n)n)n)n)n)n ≫ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| = \underline{(ϵ)})n)n)n)n; fromNotLess ⊳ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| = \underline{(ϵ)})n)n)n)n ≫ \underline{(ϵ)} <= |(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))|; A4 @ \underline{(n2)} ⊳ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n)n)n ≫ ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{(n2)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= (rec(1+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}]+(−u\underline{(fx)}[\underline{(v2)}]))| = (rec(1+1)∗\underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n)n)n; A4 @ \underline{m} ⊳ ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{(n2)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n)n)n ≫ ˙¬ (0 <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{(n2)} ⇒ \underline{(n1)} <= \underline{m} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fx)}[\underline{m}]))| <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fx)}[\underline{m}]))| = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{m}]))| <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{m}]))| = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n)n)n; PositiveHalved ⊳ ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ≫ ˙¬ (0 <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n; leqTransitivity ⊳ \underline{(n1)} <= \underline{(n2)} ⊳ \underline{(n2)} <= \underline{m} ≫ \underline{(n1)} <= \underline{m}; MP3 ⊳ ˙¬ (0 <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{(n2)} ⇒ \underline{(n1)} <= \underline{m} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fx)}[\underline{m}]))| <= (rec(1+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}]+(−u\underline{(fx)}[\underline{m}]))| = (rec(1+1)∗\underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{m}]))| <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{m}]))| = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n)n)n ⊳ ˙¬ (0 <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n ⊳ \underline{(n1)} <= \underline{(n2)} ⊳ \underline{(n1)} <= \underline{m} ≫ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fx)}[\underline{m}]))| <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fx)}[\underline{m}]))| = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{m}]))| <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{m}]))| = (rec(1+1)∗\underline{(ϵ)}))n)n)n)n)n; FirstConjunct⊳ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}]+(−u\underline{(fx)}[\underline{m}]))| <= (rec(1+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}]+(−u\underline{(fx)}[\underline{m}]))| = (rec(1+1)∗\underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{m}]))| <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{m}]))| = (rec(1+1)∗\underline{(ϵ)}))n)n)n)n)n ≫ ˙¬ (|(\underline{(fx)}[\underline{(n2)}]+(−u\underline{(fx)}[\underline{m}]))| <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fx)}[\underline{m}]))| = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n; SecondConjunct ⊳ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fx)}[\underline{m}]))| <= (rec(1+1)∗\underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}]+(−u\underline{(fx)}[\underline{m}]))| = (rec(1+1)∗\underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{m}]))| <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{m}]))| = (rec(1+1)∗\underline{(ϵ)}))n)n)n)n)n ≫ ˙¬ (|(\underline{(fy)}[\underline{(n2)}]+(−u\underline{(fy)}[\underline{m}]))| <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{m}]))| = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n; FromNotSameF(Strong)(Helper2) ⊳ \underline{(ϵ)} <= |(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| ⊳ ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fx)}[\underline{m}]))| <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fx)}[\underline{m}]))| = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n ⊳ ˙¬ (|(\underline{(fy)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{m}]))| <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{m}]))| = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n ≫ ˙¬ (\underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}])n; ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{m}: ∀\underline{(n1)}: ∀\underline{(n2)}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: Ded ⊳ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{m}: ∀\underline{(n1)}: ∀\underline{(n2)}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| = \underline{(ϵ)})n)n)n)n ⊢ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n)n)n ⊢ \underline{(n2)} <= \underline{m} ⊢ ˙¬ (\underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}])n ≫ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| = \underline{(ϵ)})n)n)n)n ⇒ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n)n)n ⇒ \underline{(n2)} <= \underline{m} ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}])n⌉, p_{0}, c)]

[FromNotSameF(Strong)(Helper) →^{stmt} SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{m}: ∀\underline{(n1)}: ∀\underline{(n2)}: ∀\underline{(ϵ)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| = \underline{(ϵ)})n)n)n)n ⇒ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n)n)n ⇒ \underline{(n2)} <= \underline{m} ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}])n]

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

[FromNotSameF(Strong)(Helper) →^{pyk} “lemma fromNotSameF(Strong) helper”]

FromNotSameF(Strong)

[FromNotSameF(Strong) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{m}: ∀\underline{n}: ∀\underline{(n1)}: ∀\underline{(n2)}: ∀\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 ⊢ 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; 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}] + (−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}\underline{(ϵ)}: ˙¬ (∀_{obj}\underline{(n1)}: ˙¬ (∀_{obj}\underline{(n2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(n2)}]+(−u\underline{(fy)}[\underline{(n2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}]+(−u\underline{(fy)}[\underline{(n2)}]))| = \underline{(ϵ)})n)n)n)n)n)n; AEA − negated ⊳ ˙¬ (∀_{obj}\underline{(ϵ)}: ˙¬ (∀_{obj}\underline{(n1)}: ˙¬ (∀_{obj}\underline{(n2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(n2)}]+(−u\underline{(fy)}[\underline{(n2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}]+(−u\underline{(fy)}[\underline{(n2)}]))| = \underline{(ϵ)})n)n)n)n)n)n ≫ ˙¬ (∀_{obj}\underline{(ϵ)}: ˙¬ (∀_{obj}\underline{(n1)}: ˙¬ (∀_{obj}\underline{(n2)}: ˙¬ ( ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| = \underline{(ϵ)})n)n)n)n)n)n)n)n; Cauchy(2) ≫ ∀_{obj}\underline{(ϵ)}: ˙¬ (∀_{obj}\underline{(n1)}: ˙¬ (∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n)n)n)n)n; A4 @(rec(1 + 1) ∗ \underline{(ϵ)}) ⊳ ∀_{obj}\underline{(ϵ)}: ˙¬ (∀_{obj}\underline{(n1)}: ˙¬ (∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = \underline{(ϵ)})n)n)n)n)n)n)n ≫ ˙¬ (∀_{obj}\underline{(n1)}: ˙¬ (∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n)n)n)n)n; FromNotSameF(Strong)(Helper) ≫ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| = \underline{(ϵ)})n)n)n)n ⇒ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n)n)n ⇒ \underline{(n2)} <= \underline{m} ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}])n; EAE − MP ⊳ ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}] + (−u\underline{(fy)}[\underline{(n2)}]))| = \underline{(ϵ)})n)n)n)n ⇒ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n)n)n ⇒ \underline{(n2)} <= \underline{m} ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}])n ⊳ ˙¬ (∀_{obj}\underline{(ϵ)}: ˙¬ (∀_{obj}\underline{(n1)}: ˙¬ (∀_{obj}\underline{(n2)}: ˙¬ ( ˙¬ ( ˙¬ (0 <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \underline{(ϵ)})n)n)n ⇒ \underline{(n1)} <= \underline{(n2)} ⇒ ˙¬ (|(\underline{(fx)}[\underline{(n2)}]+(−u\underline{(fy)}[\underline{(n2)}]))| <= \underline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(n2)}]+(−u\underline{(fy)}[\underline{(n2)}]))| = \underline{(ϵ)})n)n)n)n)n)n)n)n ≫ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n)n)n ⇒ \underline{(n2)} <= \underline{m} ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}])n; ExistMP ⊳ ∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n)n)n ⇒ \underline{(n2)} <= \underline{m} ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}])n ⊳ ˙¬ (∀_{obj}\underline{(n1)}: ˙¬ (∀_{obj}\underline{(v1)}: ∀_{obj}\underline{(v2)}: ˙¬ (0 <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ \underline{(n1)} <= \underline{(v1)} ⇒ \underline{(n1)} <= \underline{(v2)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\underline{(v1)}] + (−u\underline{(fx)}[\underline{(v2)}]))| = (rec(1 + 1) ∗ \underline{(ϵ)}))n)n)n ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| <= (rec(1 + 1) ∗ \underline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fy)}[\underline{(v1)}] + (−u\underline{(fy)}[\underline{(v2)}]))| = (rec(1+1)∗\underline{(ϵ)}))n)n)n)n)n)n)n ≫ \underline{(n2)} <= \underline{m} ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}])n; Gen ⊳ \underline{(n2)} <= \underline{m} ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}])n ≫ ∀_{obj}\underline{m}: \underline{(n2)} <= \underline{m} ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}])n; IntroExist @ \underline{(n2)} ⊳ ∀_{obj}\underline{m}: \underline{(n2)} <= \underline{m} ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}])n ≫ ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{m}: \underline{n} <= \underline{m} ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}])n)n)n⌉, p_{0}, c)]

[FromNotSameF(Strong) →^{stmt} SystemQ ⊢ ∀\underline{(v1)}: ∀\underline{(v2)}: ∀\underline{m}: ∀\underline{n}: ∀\underline{(n1)}: ∀\underline{(n2)}: ∀\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 ⊢ ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{m}: \underline{n} <= \underline{m} ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}])n)n)n]

[FromNotSameF(Strong) →^{tex} “FromNotSameF(Strong)”]

[FromNotSameF(Strong) →^{pyk} “lemma fromNotSameF(Strong)”]

SameFreciprocal(Helper)

[SameFreciprocal(Helper) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: ∀\underline{(fx)}: ∀_{obj}\underline{m}: \underline{n} <= \underline{m} ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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 ⊢ \underline{n} <= \underline{m} ⊢ A4 @ \underline{m} ⊳ ∀_{obj}\underline{m}: \underline{n} <= \underline{m} ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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 ≫ \underline{n} <= \underline{m} ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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; MP ⊳ \underline{n} <= \underline{m} ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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 ⊳ \underline{n} <= \underline{m} ≫ ˙¬ (\underline{(fx)}[\underline{m}] = \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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; NatType ≫ \underline{m} ∈ N; 0f ⊳ \underline{m} ∈ N ≫ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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}] = 0; SubNeqRight ⊳ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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}] = 0 ⊳ ˙¬ (\underline{(fx)}[\underline{m}] = {ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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 ≫ ˙¬ (\underline{(fx)}[\underline{m}] = 0)n; ReciprocalFnonzero ⊳ ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ≫ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}] = rec\underline{(fx)}[\underline{m}]; EqMultiplicationLeft ⊳ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}] = rec\underline{(fx)}[\underline{m}] ≫ (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}]) = (\underline{(fx)}[\underline{m}]∗rec\underline{(fx)}[\underline{m}]); Reciprocal⊳ ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ≫ (\underline{(fx)}[\underline{m}] ∗ rec\underline{(fx)}[\underline{m}]) = 1; 1f ⊳ \underline{m} ∈ N ≫ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] = 1; eqSymmetry ⊳ {ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] = 1 ≫ 1 = \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}]; eqTransitivity ⊳ (\underline{(fx)}[\underline{m}] ∗ rec\underline{(fx)}[\underline{m}]) = 1 ⊳ 1 = \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] ≫ (\underline{(fx)}[\underline{m}] ∗ rec\underline{(fx)}[\underline{m}]) = \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}]; TimesF ≫ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}]); eqTransitivity4 ⊳ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}])\}\})n)n\}[\underline{m}] = (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}]) ⊳ (\underline{(fx)}[\underline{m}] ∗ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}]) = (\underline{(fx)}[\underline{m}] ∗ rec\underline{(fx)}[\underline{m}]) ⊳ (\underline{(fx)}[\underline{m}] ∗ rec\underline{(fx)}[\underline{m}]) = {ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}: ˙¬ (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}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)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}]; ∀\underline{m}: ∀\underline{n}: ∀\underline{(fx)}: Ded ⊳ ∀\underline{m}: ∀\underline{n}: ∀\underline{(fx)}: ∀_{obj}\underline{m}: \underline{n} <= \underline{m} ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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 ⊢ \underline{n} <= \underline{m} ⊢ {ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)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}] ≫ ∀_{obj}\underline{m}: \underline{n} <= \underline{m} ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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 ⇒ \underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)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}]⌉, p_{0}, c)]

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

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

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

SameFreciprocal

[SameFreciprocal →^{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}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n)n ⊢ FromNotSameF(Strong) ⊳ ˙¬ (∀_{obj}\overline{(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\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, 0\}\})n)n\}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n)n ≫ ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{m}: \underline{n} <= \underline{m} ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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)n; SameFreciprocal(Helper) ≫ ∀_{obj}\underline{m}: \underline{n} <= \underline{m} ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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 ⇒ \underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)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}]; ExistMP ⊳ ∀_{obj}\underline{m}: \underline{n} <= \underline{m} ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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 ⇒ \underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)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}] ⊳ ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{m}: \underline{n} <= \underline{m} ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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)n ≫ \underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)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}]; Gen ⊳ \underline{n} <= \underline{m} ⇒ {ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)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}] ≫ ∀_{obj}\underline{m}: \underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)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}]; IntroExist @ \underline{n} ⊳ \underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)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}] ≫ ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{m}: \underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)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; (Eventually = f)2sameF ⊳ ˙¬ (∀_{obj}\underline{n}: ˙¬ (∀_{obj}\underline{m}: \underline{n} <= \underline{m} ⇒ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)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 ≫ ∀_{obj}\overline{(ϵ)}: ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{n} <= \overline{m} ⇒ ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|({ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}[\underline{m}])\}\})n)n\}[\overline{m}] + (−u\{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}]))| = \overline{(ϵ)})n)n)n)n)n⌉, p_{0}, c)]

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

[SameFreciprocal →^{tex} “SameFreciprocal”]

[SameFreciprocal →^{pyk} “lemma sameFreciprocal”]

From!! ==

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

[From!! ==→^{tex} “From!!==”]

[From!! == →^{pyk} “lemma from!!==”]

Reciprocal(R)

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

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

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

[Reciprocal(R) →^{pyk} “lemma reciprocal(R)”]

TimesCommutativity(F)

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

[TimesCommutativity(F) →^{pyk} “lemma timesCommutativity(F)”]

Distribution(F)

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

[Distribution(F) →^{pyk} “lemma distribution(F)”]

FromMax(1)

[FromMax(1) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: \underline{y} <= \underline{x} ⊢ Max ≫ ˙¬ ( ˙¬ (\underline{y} <= \underline{x} ⇒ ˙¬ (if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) = \underline{x})n)n)n ⇒ ˙¬ ( ˙¬ (\underline{y} <= \underline{x})n ⇒ ˙¬ (if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) = \underline{y})n)n; AddDoubleNeg ⊳ \underline{y} <= \underline{x} ≫ ˙¬ ( ˙¬ (\underline{y} <= \underline{x})n)n; ToNegatedAnd(1) ⊳ ˙¬ ( ˙¬ (\underline{y} <= \underline{x})n)n ≫ ˙¬ ( ˙¬ ( ˙¬ (\underline{y} <= \underline{x})n ⇒ ˙¬ (if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) = \underline{y})n)n)n; NegateDisjunct2 ⊳ ˙¬ ( ˙¬ (\underline{y} <= \underline{x} ⇒ ˙¬ (if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) = \underline{x})n)n)n ⇒ ˙¬ ( ˙¬ (\underline{y} <= \underline{x})n ⇒ ˙¬ (if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) = \underline{y})n)n ⊳ ˙¬ ( ˙¬ ( ˙¬ (\underline{y} <= \underline{x})n ⇒ ˙¬ (if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) = \underline{y})n)n)n ≫ ˙¬ (\underline{y} <= \underline{x} ⇒ ˙¬ (if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) = \underline{x})n)n; SecondConjunct ⊳ ˙¬ (\underline{y} <= \underline{x} ⇒ ˙¬ (if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) = \underline{x})n)n ≫ if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) = \underline{x}⌉, p_{0}, c)]

[FromMax(1) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: \underline{y} <= \underline{x} ⊢ if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) = \underline{x}]

[FromMax(1) →^{tex} “FromMax(1)”]

[FromMax(1) →^{pyk} “lemma fromMax(1)”]

FromMax(2)

[FromMax(2) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ (\underline{y} <= \underline{x})n ⊢ Max ≫ ˙¬ ( ˙¬ (\underline{y} <= \underline{x} ⇒ ˙¬ (if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) = \underline{x})n)n)n ⇒ ˙¬ ( ˙¬ (\underline{y} <= \underline{x})n ⇒ ˙¬ (if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) = \underline{y})n)n; ToNegatedAnd(1) ⊳ ˙¬ (\underline{y} <= \underline{x})n ≫ ˙¬ ( ˙¬ (\underline{y} <= \underline{x} ⇒ ˙¬ (if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) = \underline{x})n)n)n; NegateDisjunct1 ⊳ ˙¬ ( ˙¬ (\underline{y} <= \underline{x} ⇒ ˙¬ (if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) = \underline{x})n)n)n ⇒ ˙¬ ( ˙¬ (\underline{y} <= \underline{x})n ⇒ ˙¬ (if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) = \underline{y})n)n ⊳ ˙¬ ( ˙¬ (\underline{y} <= \underline{x} ⇒ ˙¬ (if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) = \underline{x})n)n)n ≫ ˙¬ ( ˙¬ (\underline{y} <= \underline{x})n ⇒ ˙¬ (if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) = \underline{y})n)n; SecondConjunct ⊳ ˙¬ ( ˙¬ (\underline{y} <= \underline{x})n ⇒ ˙¬ (if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) = \underline{y})n)n ≫ if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) = \underline{y}⌉, p_{0}, c)]

[FromMax(2) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ (\underline{y} <= \underline{x})n ⊢ if(\underline{y} <= \underline{x}, \underline{x}, \underline{y}) = \underline{y}] [FromMax(2) →^{tex} “FromMax(2)”]

[FromMax(2) →^{pyk} “lemma fromMax(2)”]

ToNegatedAnd

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

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

[ToNegatedAnd →^{tex} “ToNegatedAnd”]

[ToNegatedAnd →^{pyk} “prop lemma to negated and”]

DistributionOut

[DistributionOut →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: Distribution ≫ (\underline{x} ∗ (\underline{y} + \underline{z})) = ((\underline{x} ∗ \underline{y}) + (\underline{x} ∗ \underline{z})); eqSymmetry ⊳ (\underline{x} ∗ (\underline{y} + \underline{z})) = ((\underline{x} ∗ \underline{y}) + (\underline{x} ∗ \underline{z})) ≫ ((\underline{x} ∗ \underline{y}) + (\underline{x} ∗ \underline{z})) = (\underline{x} ∗ (\underline{y} + \underline{z}))⌉, p_{0}, c)]

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

[DistributionOut →^{tex} “DistributionOut”]

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

DistributionOutLeft

[DistributionOutLeft →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: timesCommutativity ≫ (\underline{y} ∗ \underline{x}) = (\underline{x} ∗ \underline{y}); timesCommutativity ≫ (\underline{z} ∗ \underline{x}) = (\underline{x} ∗ \underline{z}); AddEquations ⊳ (\underline{y} ∗ \underline{x}) = (\underline{x} ∗ \underline{y}) ⊳ (\underline{z} ∗ \underline{x}) = (\underline{x} ∗ \underline{z}) ≫ ((\underline{y} ∗ \underline{x}) + (\underline{z} ∗ \underline{x})) = ((\underline{x} ∗ \underline{y}) + (\underline{x} ∗ \underline{z})); DistributionOut ≫ ((\underline{x} ∗ \underline{y}) + (\underline{x} ∗ \underline{z})) = (\underline{x}∗(\underline{y}+\underline{z})); eqTransitivity⊳((\underline{y}∗\underline{x})+(\underline{z}∗\underline{x})) = ((\underline{x}∗\underline{y})+(\underline{x}∗\underline{z}))⊳((\underline{x}∗\underline{y})+(\underline{x}∗\underline{z})) = (\underline{x} ∗ (\underline{y} + \underline{z})) ≫ ((\underline{y} ∗ \underline{x}) + (\underline{z} ∗ \underline{x})) = (\underline{x} ∗ (\underline{y} + \underline{z}))⌉, p_{0}, c)]

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

[DistributionOutLeft →^{tex} “DistributionOutLeft”]

[DistributionOutLeft →^{pyk} “lemma distributionOutLeft”]

DistributionLeft

[DistributionLeft →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: DistributionOutLeft ≫ ((\underline{y} ∗ \underline{x}) + (\underline{z} ∗ \underline{x})) = (\underline{x} ∗ (\underline{y} + \underline{z})); eqSymmetry ⊳ ((\underline{y} ∗ \underline{x}) + (\underline{z} ∗ \underline{x})) = (\underline{x} ∗ (\underline{y} + \underline{z})) ≫ (\underline{x} ∗ (\underline{y} + \underline{z})) = ((\underline{y} ∗ \underline{x}) + (\underline{z} ∗ \underline{x}))⌉, p_{0}, c)]

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

[DistributionLeft →^{tex} “DistributionLeft”]

[DistributionLeft →^{pyk} “lemma distributionLeft”]

FromNotLess(R)

[FromNotLess(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{(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; ∀\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{(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\})n ⇒ ˙¬ (∀_{obj}\overline{(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 ⊢ NegativeMT ⊳ ˙¬ ( ˙¬ ( ˙¬ (∀_{obj}\overline{(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{(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)]

[FromNotLess(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 ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(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\}]

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

[FromNotLess(R) →^{pyk} “lemma fromNotLess(R)”]

CartProdIsRelation

[CartProdIsRelation →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sy)}: \overline{(r1)} ∈ \{ph ∈ P(P(Union(\{\underline{(sx)}, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ⊢ Sep2Formula ⊳ \overline{(r1)} ∈ \{ph ∈ P(P(Union(\{\underline{(sx)}, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ≫ ˙¬ (\overline{(r1)} ∈ P(P(Union(\{\underline{(sx)}, \underline{(sy)}\}))) ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n)n)n; SecondConjunct ⊳ ˙¬ (\overline{(r1)} ∈ P(P(Union(\{\underline{(sx)}, \underline{(sy)}\}))) ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n)n)n ≫ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n; ∀\underline{(sx)}: ∀\underline{(sy)}: Ded ⊳ ∀\underline{(sx)}: ∀\underline{(sy)}: \overline{(r1)} ∈ \{ph ∈ P(P(Union(\{\underline{(sx)}, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ⊢ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ≫ \overline{(r1)} ∈ \{ph ∈ P(P(Union(\{\underline{(sx)}, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n; Gen ⊳ \overline{(r1)} ∈ \{ph ∈ P(P(Union(\{\underline{(sx)}, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ≫ ∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \{ph ∈ P(P(Union(\{\underline{(sx)}, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n; Repetition ⊳ ∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \{ph ∈ P(P(Union(\{\underline{(sx)}, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ≫ ∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \{ph ∈ P(P(Union(\{\underline{(sx)}, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n⌉, p_{0}, c)] [CartProdIsRelation →^{stmt} SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sy)}: ∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \{ph ∈ P(P(Union(\{\underline{(sx)}, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n] [CartProdIsRelation →^{tex} “CartProdIsRelation”]

[CartProdIsRelation →^{pyk} “lemma cartProdIsRelation”]

FromSubset

[FromSubset →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sy)}: ∀\underline{(sz)}: ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \underline{(sx)} ⇒ \overline{(s1)} ∈ \underline{(sy)} ⊢ \underline{(sz)} ∈ \underline{(sx)} ⊢ Repetition ⊳ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \underline{(sx)} ⇒ \overline{(s1)} ∈ \underline{(sy)} ≫ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \underline{(sx)} ⇒ \overline{(s1)} ∈ \underline{(sy)}; A4 @ \underline{(sz)} ⊳ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \underline{(sx)} ⇒ \overline{(s1)} ∈ \underline{(sy)} ≫ \underline{(sz)} ∈ \underline{(sx)} ⇒ \underline{(sz)} ∈ \underline{(sy)}; MP ⊳ \underline{(sz)} ∈ \underline{(sx)} ⇒ \underline{(sz)} ∈ \underline{(sy)} ⊳ \underline{(sz)} ∈ \underline{(sx)} ≫ \underline{(sz)} ∈ \underline{(sy)}⌉, p_{0}, c)]

[FromSubset →^{stmt} SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sy)}: ∀\underline{(sz)}: ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \underline{(sx)} ⇒ \overline{(s1)} ∈ \underline{(sy)} ⊢ \underline{(sz)} ∈ \underline{(sx)} ⊢ \underline{(sz)} ∈ \underline{(sy)}]

[FromSubset →^{tex} “FromSubset”]

[FromSubset →^{pyk} “lemma fromSubset”]

SubsetIsRelation

[SubsetIsRelation →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sy)}: ∀\underline{(sz)}: ∀\underline{(su)}: ∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(sx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sz)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(su)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⊢ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \underline{(sy)} ⇒ \overline{(s1)} ∈ \underline{(sx)} ⊢ \overline{(r1)} ∈ \underline{(sy)} ⊢ Repetition ⊳ ∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(sx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sz)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(su)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ≫ ∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(sx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sz)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(su)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n; A4 @ \overline{(r1)}⊳∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(sx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sz)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(su)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ≫ \overline{(r1)} ∈ \underline{(sx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sz)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(su)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n; FromSubset ⊳ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \underline{(sy)} ⇒ \overline{(s1)} ∈ \underline{(sx)} ⊳ \overline{(r1)} ∈ \underline{(sy)} ≫ \overline{(r1)} ∈ \underline{(sx)}; MP ⊳ \overline{(r1)} ∈ \underline{(sx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sz)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(su)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⊳ \overline{(r1)} ∈ \underline{(sx)} ≫ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sz)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(su)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n; ∀\underline{(sx)}: ∀\underline{(sy)}: ∀\underline{(sz)}: ∀\underline{(su)}: Ded ⊳ ∀\underline{(sx)}: ∀\underline{(sy)}: ∀\underline{(sz)}: ∀\underline{(su)}: ∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(sx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sz)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(su)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⊢ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \underline{(sy)} ⇒ \overline{(s1)} ∈ \underline{(sx)} ⊢ \overline{(r1)} ∈ \underline{(sy)} ⊢ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sz)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(su)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ≫ ∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(sx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sz)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(su)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \underline{(sy)} ⇒ \overline{(s1)} ∈ \underline{(sx)} ⇒ \overline{(r1)} ∈ \underline{(sy)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sz)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(su)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n; ∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(sx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sz)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(su)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⊢ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \underline{(sy)} ⇒ \overline{(s1)} ∈ \underline{(sx)} ⊢ MP2 ⊳ ∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(sx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sz)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(su)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \underline{(sy)} ⇒ \overline{(s1)} ∈ \underline{(sx)} ⇒ \overline{(r1)} ∈ \underline{(sy)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sz)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(su)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⊳ ∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(sx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sz)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(su)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⊳ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \underline{(sy)} ⇒ \overline{(s1)} ∈ \underline{(sx)} ≫ \overline{(r1)} ∈ \underline{(sy)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sz)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(su)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n; Gen ⊳ \overline{(r1)} ∈ \underline{(sy)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sz)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(su)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ≫ ∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(sy)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sz)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(su)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n; Repetition ⊳ ∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(sy)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sz)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(su)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ≫ ∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(sy)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sz)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(su)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n⌉, p_{0}, c)] [SubsetIsRelation →^{stmt} SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sy)}: ∀\underline{(sz)}: ∀\underline{(su)}: ∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(sx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sz)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(su)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⊢ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \underline{(sy)} ⇒ \overline{(s1)} ∈ \underline{(sx)} ⊢ ∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(sy)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sz)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(su)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n] [SubsetIsRelation →^{tex} “SubsetIsRelation”]

[SubsetIsRelation →^{pyk} “lemma subsetIsRelation”]

ToSeries

[ToSeries →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(sy)}: ∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⊢ ∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)} ⊢ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n ⊢ Repetition ⊳ ∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ≫ ∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n; JoinConjuncts ⊳ ∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⊳ ∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)} ≫ ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n; Repetition ⊳ ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ≫ ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n; JoinConjuncts ⊳ ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⊳ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n; Repetition ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n⌉, p_{0}, c)]

[ToSeries →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(sy)}: ∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⊢ ∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)} ⊢ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n]

[ToSeries →^{tex} “ToSeries”]

[ToSeries →^{pyk} “lemma toSeries”]

FromSeries

[FromSeries →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(sy)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ⊢ Repetition ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n; Repetition ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n; Repetition ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n; Repetition ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n⌉, p_{0}, c)]

[FromSeries →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(sy)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n]

[FromSeries →^{tex} “FromSeries”]

[FromSeries →^{pyk} “lemma fromSeries”]

SeriesSubsetCP

[SeriesSubsetCP →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(sy)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ⊢ \overline{(s1)} ∈ \underline{(fx)} ⊢ FromSeries ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n; FirstConjunct ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ≫ ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n; FirstConjunct ⊳ ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ≫ ∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n; A4 @ \overline{(s1)} ⊳ ∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ≫ \overline{(s1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(s1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n; MP ⊳ \overline{(s1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(s1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⊳ \overline{(s1)} ∈ \underline{(fx)} ≫ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(s1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n; ∀\underline{(sy)}: ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(s1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n ⊢ FirstConjunct ⊳ ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(s1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n ≫ ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n; FirstConjunct ⊳ ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ≫ \overline{(op1)} ∈ N; SecondConjunct ⊳ ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ≫ \overline{(op2)} ∈ \underline{(sy)}; SecondConjunct ⊳ ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(s1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n ≫ \overline{(s1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\}; eqSymmetry ⊳ \overline{(s1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\} ≫ \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\} = \overline{(s1)}; ToCartProd ⊳ \overline{(op1)} ∈ N ⊳ \overline{(op2)} ∈ \underline{(sy)} ≫ \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\} ∈ \{ph ∈ P(P(Union(\{N, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}; SameMember ⊳ \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\} = \overline{(s1)} ⊳ \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\} ∈ \{ph ∈ P(P(Union(\{N, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ≫ \overline{(s1)} ∈ \{ph ∈ P(P(Union(\{N, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}; Ded ⊳ ∀\underline{(sy)}: ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(s1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n ⊢ \overline{(s1)} ∈ \{ph ∈ P(P(Union(\{N, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ≫ ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(s1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n ⇒ \overline{(s1)} ∈ \{ph ∈ P(P(Union(\{N, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}; TwiceExistMP ⊳ ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(s1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n ⇒ \overline{(s1)} ∈ \{ph ∈ P(P(Union(\{N, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ⊳ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(s1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ≫ \overline{(s1)} ∈ \{ph ∈ P(P(Union(\{N, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}; ∀\underline{(fx)}: ∀\underline{(sy)}: Ded ⊳ ∀\underline{(fx)}: ∀\underline{(sy)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ⊢ \overline{(s1)} ∈ \underline{(fx)} ⊢ \overline{(s1)} ∈ \{ph ∈ P(P(Union(\{N, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ⇒ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \underline{(fx)} ⇒ \overline{(s1)} ∈ \{ph ∈ P(P(Union(\{N, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}; ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ⊢ MP ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ⇒ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \underline{(fx)} ⇒ \overline{(s1)} ∈ \{ph ∈ P(P(Union(\{N, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ≫ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \underline{(fx)} ⇒ \overline{(s1)} ∈ \{ph ∈ P(P(Union(\{N, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}; Repetition ⊳ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \underline{(fx)} ⇒ \overline{(s1)} ∈ \{ph ∈ P(P(Union(\{N, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ≫ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \underline{(fx)} ⇒ \overline{(s1)} ∈ \{ph ∈ P(P(Union(\{N, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}⌉, p_{0}, c)] [SeriesSubsetCP →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(sy)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ⊢ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \underline{(fx)} ⇒ \overline{(s1)} ∈ \{ph ∈ P(P(Union(\{N, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}] [SeriesSubsetCP →^{tex} “SeriesSubsetCP”]

[SeriesSubsetCP →^{pyk} “lemma seriesSubsetCP”]

ValueType

[ValueType →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(sy)}: \underline{m} ∈ N ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ⊢ MemberOfSeries ⊳ \underline{m} ∈ N ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ≫ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fx)}[\underline{m}]\}\} ∈ \underline{(fx)}; SeriesSubsetCP ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ≫ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \underline{(fx)} ⇒ \overline{(s1)} ∈ \{ph ∈ P(P(Union(\{N, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}; FromSubset ⊳ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \underline{(fx)} ⇒ \overline{(s1)} ∈ \{ph ∈ P(P(Union(\{N, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ⊳ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fx)}[\underline{m}]\}\} ∈ \underline{(fx)} ≫ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fx)}[\underline{m}]\}\} ∈ \{ph ∈ P(P(Union(\{N, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}; FromCartProd(2) ⊳ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fx)}[\underline{m}]\}\} ∈ \{ph ∈ P(P(Union(\{N, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ≫ \underline{(fx)}[\underline{m}] ∈ \underline{(sy)}⌉, p_{0}, c)]

[ValueType →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(sy)}: \underline{m} ∈ N ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ⊢ \underline{(fx)}[\underline{m}] ∈ \underline{(sy)}]

[ValueType →^{tex} “ValueType”]

[ValueType →^{pyk} “lemma valueType”]

RemoveOr

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

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

[RemoveOr →^{tex} “RemoveOr”]

[RemoveOr →^{pyk} “prop lemma remove or”]

FromSingleton

[FromSingleton →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sy)}: \underline{(sx)} ∈ \{\underline{(sy)}, \underline{(sy)}\} ⊢ Repetition ⊳ \underline{(sx)} ∈ \{\underline{(sy)}, \underline{(sy)}\} ≫ \underline{(sx)} ∈ \{\underline{(sy)}, \underline{(sy)}\}; Pair2Formula ⊳ \underline{(sx)} ∈ \{\underline{(sy)}, \underline{(sy)}\} ≫ ˙¬ (\underline{(sx)} = \underline{(sy)})n ⇒ \underline{(sx)} = \underline{(sy)}; RemoveOr ⊳ ˙¬ (\underline{(sx)} = \underline{(sy)})n ⇒ \underline{(sx)} = \underline{(sy)} ≫ \underline{(sx)} = \underline{(sy)}⌉, p_{0}, c)]

[FromSingleton →^{stmt} SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sy)}: \underline{(sx)} ∈ \{\underline{(sy)}, \underline{(sy)}\} ⊢ \underline{(sx)} = \underline{(sy)}]

[FromSingleton →^{tex} “FromSingleton”]

[FromSingleton →^{pyk} “lemma fromSingleton”]

InPair(1)

[InPair(1) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sy)}: eqReflexivity ≫ \underline{(sx)} = \underline{(sx)}; WeakenOr2 ⊳ \underline{(sx)} = \underline{(sx)} ≫ ˙¬ (\underline{(sx)} = \underline{(sx)})n ⇒ \underline{(sx)} = \underline{(sy)}; Formula2Pair ⊳ ˙¬ (\underline{(sx)} = \underline{(sx)})n ⇒ \underline{(sx)} = \underline{(sy)} ≫ \underline{(sx)} ∈ \{\underline{(sx)}, \underline{(sy)}\}⌉, p_{0}, c)]

[InPair(1) →^{stmt} SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sy)}: \underline{(sx)} ∈ \{\underline{(sx)}, \underline{(sy)}\}] [InPair(1) →^{tex} “InPair(1)”]

[InPair(1) →^{pyk} “lemma inPair(1)”]

InPair(2)

[InPair(2) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sy)}: eqReflexivity ≫ \underline{(sy)} = \underline{(sy)}; WeakenOr1 ⊳ \underline{(sy)} = \underline{(sy)} ≫ ˙¬ (\underline{(sy)} = \underline{(sx)})n ⇒ \underline{(sy)} = \underline{(sy)}; Formula2Pair ⊳ ˙¬ (\underline{(sy)} = \underline{(sx)})n ⇒ \underline{(sy)} = \underline{(sy)} ≫ \underline{(sy)} ∈ \{\underline{(sx)}, \underline{(sy)}\}⌉, p_{0}, c)]

[InPair(2) →^{stmt} SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sy)}: \underline{(sy)} ∈ \{\underline{(sx)}, \underline{(sy)}\}]

[InPair(2) →^{tex} “InPair(2)”]

[InPair(2) →^{pyk} “lemma inPair(2)”]

SameMember(2)

[SameMember(2) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sy)}: ∀\underline{(sz)}: \underline{(sx)} = \underline{(sy)} ⊢ \underline{(sy)} ∈ \underline{(sz)} ⊢ eqSymmetry ⊳ \underline{(sx)} = \underline{(sy)} ≫ \underline{(sy)} = \underline{(sx)}; SameMember ⊳ \underline{(sy)} = \underline{(sx)} ⊳ \underline{(sy)} ∈ \underline{(sz)} ≫ \underline{(sx)} ∈ \underline{(sz)}⌉, p_{0}, c)]

[SameMember(2) →^{stmt} SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sy)}: ∀\underline{(sz)}: \underline{(sx)} = \underline{(sy)} ⊢ \underline{(sy)} ∈ \underline{(sz)} ⊢ \underline{(sx)} ∈ \underline{(sz)}]

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

[SameMember(2) →^{pyk} “lemma sameMember(2)”]

ToBinaryUnion(1)

[ToBinaryUnion(1) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sy)}: ∀\underline{(sz)}: ∀\underline{(su)}: \underline{(sx)} ∈ \underline{(sy)} ⊢ InPair(1) ≫ \underline{(sy)} ∈ \{\underline{(sy)}, \underline{(sz)}\}; JoinConjuncts ⊳ \underline{(sx)} ∈ \underline{(sy)} ⊳ \underline{(sy)} ∈ \{\underline{(sy)}, \underline{(sz)}\} ≫ ˙¬ (\underline{(sx)} ∈ \underline{(sy)} ⇒ ˙¬ (\underline{(sy)} ∈ \{\underline{(sy)}, \underline{(sz)}\})n)n; IntroExist @ \underline{(sy)} ⊳ ˙¬ (\underline{(sx)} ∈ \underline{(sy)} ⇒ ˙¬ (\underline{(sy)} ∈ \{\underline{(sy)}, \underline{(sz)}\})n)n ≫ ˙¬ (∀_{obj}\underline{(su)}: ˙¬ ( ˙¬ (\underline{(sx)} ∈ \underline{(su)} ⇒ ˙¬ (\underline{(su)} ∈ \{\underline{(sy)}, \underline{(sz)}\})n)n)n)n; Formula2Union ⊳ ˙¬ (∀_{obj}\underline{(su)}: ˙¬ ( ˙¬ (\underline{(sx)} ∈ \underline{(su)} ⇒ ˙¬ (\underline{(su)} ∈ \{\underline{(sy)}, \underline{(sz)}\})n)n)n)n ≫ \underline{(sx)} ∈ Union(\{\underline{(sy)}, \underline{(sz)}\}); Repetition ⊳ \underline{(sx)} ∈ Union(\{\underline{(sy)}, \underline{(sz)}\}) ≫ \underline{(sx)} ∈ Union(\{\underline{(sy)}, \underline{(sz)}\})⌉, p_{0}, c)]

[ToBinaryUnion(1) →^{stmt} SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sy)}: ∀\underline{(sz)}: ∀\underline{(su)}: \underline{(sx)} ∈ \underline{(sy)} ⊢ \underline{(sx)} ∈ Union(\{\underline{(sy)}, \underline{(sz)}\})] [ToBinaryUnion(1) →^{tex} “ToBinaryUnion(1)”]

[ToBinaryUnion(1) →^{pyk} “lemma toBinaryUnion(1)”]

ToBinaryUnion(2)

[ToBinaryUnion(2) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sy)}: ∀\underline{(sz)}: ∀\underline{(su)}: \underline{(sx)} ∈ \underline{(sz)} ⊢ InPair(2) ≫ \underline{(sz)} ∈ \{\underline{(sy)}, \underline{(sz)}\}; JoinConjuncts ⊳ \underline{(sx)} ∈ \underline{(sz)} ⊳ \underline{(sz)} ∈ \{\underline{(sy)}, \underline{(sz)}\} ≫ ˙¬ (\underline{(sx)} ∈ \underline{(sz)} ⇒ ˙¬ (\underline{(sz)} ∈ \{\underline{(sy)}, \underline{(sz)}\})n)n; IntroExist @ \underline{(sz)} ⊳ ˙¬ (\underline{(sx)} ∈ \underline{(sz)} ⇒ ˙¬ (\underline{(sz)} ∈ \{\underline{(sy)}, \underline{(sz)}\})n)n ≫ ˙¬ (∀_{obj}\underline{(su)}: ˙¬ ( ˙¬ (\underline{(sx)} ∈ \underline{(su)} ⇒ ˙¬ (\underline{(su)} ∈ \{\underline{(sy)}, \underline{(sz)}\})n)n)n)n; Formula2Union ⊳ ˙¬ (∀_{obj}\underline{(su)}: ˙¬ ( ˙¬ (\underline{(sx)} ∈ \underline{(su)} ⇒ ˙¬ (\underline{(su)} ∈ \{\underline{(sy)}, \underline{(sz)}\})n)n)n)n ≫ \underline{(sx)} ∈ Union(\{\underline{(sy)}, \underline{(sz)}\}); Repetition ⊳ \underline{(sx)} ∈ Union(\{\underline{(sy)}, \underline{(sz)}\}) ≫ \underline{(sx)} ∈ Union(\{\underline{(sy)}, \underline{(sz)}\})⌉, p_{0}, c)]

[ToBinaryUnion(2) →^{stmt} SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sy)}: ∀\underline{(sz)}: ∀\underline{(su)}: \underline{(sx)} ∈ \underline{(sz)} ⊢ \underline{(sx)} ∈ Union(\{\underline{(sy)}, \underline{(sz)}\})]

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

[ToBinaryUnion(2) →^{pyk} “lemma toBinaryUnion(2)”]

FromOrderedPair(TwoLevels)

[FromOrderedPair(TwoLevels) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sy)}: ∀\underline{(sz)}: ∀\underline{(su)}: \underline{(sx)} ∈ \underline{(sy)} ⊢ \underline{(sy)} ∈ \{\{\underline{(sz)}, \underline{(sz)}\}, \{\underline{(sz)}, \underline{(su)}\}\} ⊢ Repetition ⊳ \underline{(sy)} ∈ \{\{\underline{(sz)}, \underline{(sz)}\}, \{\underline{(sz)}, \underline{(su)}\}\} ≫ \underline{(sy)} ∈ \{\{\underline{(sz)}, \underline{(sz)}\}, \{\underline{(sz)}, \underline{(su)}\}\}; Pair2Formula ⊳ \underline{(sy)} ∈ \{\{\underline{(sz)}, \underline{(sz)}\}, \{\underline{(sz)}, \underline{(su)}\}\} ≫ ˙¬ (\underline{(sy)} = \{\underline{(sz)}, \underline{(sz)}\})n ⇒ \underline{(sy)} = \{\underline{(sz)}, \underline{(su)}\}; ∀\underline{(sx)}: ∀\underline{(sy)}: ∀\underline{(sz)}: ∀\underline{(su)}: \underline{(sy)} = \{\underline{(sz)}, \underline{(sz)}\} ⊢ \underline{(sx)} ∈ \underline{(sy)} ⊢ SENC1 ⊳ \underline{(sy)} = \{\underline{(sz)}, \underline{(sz)}\} ⊳ \underline{(sx)} ∈ \underline{(sy)} ≫ \underline{(sx)} ∈ \{\underline{(sz)}, \underline{(sz)}\}; FromSingleton ⊳ \underline{(sx)} ∈ \{\underline{(sz)}, \underline{(sz)}\} ≫ \underline{(sx)} = \underline{(sz)}; WeakenOr2 ⊳ \underline{(sx)} = \underline{(sz)} ≫ ˙¬ (\underline{(sx)} = \underline{(sz)})n ⇒ \underline{(sx)} = \underline{(su)}; ∀\underline{(sx)}: ∀\underline{(sy)}: ∀\underline{(sz)}: ∀\underline{(su)}: \underline{(sy)} = \{\underline{(sz)}, \underline{(su)}\} ⊢ \underline{(sx)} ∈ \underline{(sy)} ⊢ SENC1 ⊳ \underline{(sy)} = \{\underline{(sz)}, \underline{(su)}\} ⊳ \underline{(sx)} ∈ \underline{(sy)} ≫ \underline{(sx)} ∈ \{\underline{(sz)}, \underline{(su)}\}; Pair2Formula ⊳ \underline{(sx)} ∈ \{\underline{(sz)}, \underline{(su)}\} ≫ ˙¬ (\underline{(sx)} = \underline{(sz)})n ⇒ \underline{(sx)} = \underline{(su)}; Ded ⊳ ∀\underline{(sx)}: ∀\underline{(sy)}: ∀\underline{(sz)}: ∀\underline{(su)}: \underline{(sy)} = \{\underline{(sz)}, \underline{(sz)}\} ⊢ \underline{(sx)} ∈ \underline{(sy)} ⊢ ˙¬ (\underline{(sx)} = \underline{(sz)})n ⇒ \underline{(sx)} = \underline{(su)} ≫ \underline{(sy)} = \{\underline{(sz)}, \underline{(sz)}\} ⇒ \underline{(sx)} ∈ \underline{(sy)} ⇒ ˙¬ (\underline{(sx)} = \underline{(sz)})n ⇒ \underline{(sx)} = \underline{(su)}; Ded ⊳ ∀\underline{(sx)}: ∀\underline{(sy)}: ∀\underline{(sz)}: ∀\underline{(su)}: \underline{(sy)} = \{\underline{(sz)}, \underline{(su)}\} ⊢ \underline{(sx)} ∈ \underline{(sy)} ⊢ ˙¬ (\underline{(sx)} = \underline{(sz)})n ⇒ \underline{(sx)} = \underline{(su)} ≫ \underline{(sy)} = \{\underline{(sz)}, \underline{(su)}\} ⇒ \underline{(sx)} ∈ \underline{(sy)} ⇒ ˙¬ (\underline{(sx)} = \underline{(sz)})n ⇒ \underline{(sx)} = \underline{(su)}; FromDisjuncts ⊳ ˙¬ (\underline{(sy)} = \{\underline{(sz)}, \underline{(sz)}\})n ⇒ \underline{(sy)} = \{\underline{(sz)}, \underline{(su)}\} ⊳ \underline{(sy)} = \{\underline{(sz)}, \underline{(sz)}\} ⇒ \underline{(sx)} ∈ \underline{(sy)} ⇒ ˙¬ (\underline{(sx)} = \underline{(sz)})n ⇒ \underline{(sx)} = \underline{(su)} ⊳ \underline{(sy)} = \{\underline{(sz)}, \underline{(su)}\} ⇒ \underline{(sx)} ∈ \underline{(sy)} ⇒ ˙¬ (\underline{(sx)} = \underline{(sz)})n ⇒ \underline{(sx)} = \underline{(su)} ≫ \underline{(sx)} ∈ \underline{(sy)} ⇒ ˙¬ (\underline{(sx)} = \underline{(sz)})n ⇒ \underline{(sx)} = \underline{(su)}; MP ⊳ \underline{(sx)} ∈ \underline{(sy)} ⇒ ˙¬ (\underline{(sx)} = \underline{(sz)})n ⇒ \underline{(sx)} = \underline{(su)} ⊳ \underline{(sx)} ∈ \underline{(sy)} ≫ ˙¬ (\underline{(sx)} = \underline{(sz)})n ⇒ \underline{(sx)} = \underline{(su)}⌉, p_{0}, c)]

[FromOrderedPair(TwoLevels) →^{stmt} SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sy)}: ∀\underline{(sz)}: ∀\underline{(su)}: \underline{(sx)} ∈ \underline{(sy)} ⊢ \underline{(sy)} ∈ \{\{\underline{(sz)}, \underline{(sz)}\}, \{\underline{(sz)}, \underline{(su)}\}\} ⊢ ˙¬ (\underline{(sx)} = \underline{(sz)})n ⇒ \underline{(sx)} = \underline{(su)}]

[FromOrderedPair(TwoLevels) →^{tex} “FromOrderedPair(TwoLevels)”]

[FromOrderedPair(TwoLevels) →^{pyk} “lemma fromOrderedPair(twoLevels)”]

ToCartProd(Helper)

[ToCartProd(Helper) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sx1)}: ∀\underline{(sy)}: ∀\underline{(sy1)}: ∀\underline{(sz)}: \underline{(sx)} ∈ \underline{(sx1)} ⊢ \underline{(sy)} ∈ \underline{(sy1)} ⊢ \underline{(sz)} ∈ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ⊢ \overline{(s1)} ∈ \underline{(sz)} ⊢ FromOrderedPair(TwoLevels)⊳\overline{(s1)} ∈ \underline{(sz)} ⊳ \underline{(sz)} ∈ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ≫ ˙¬ (\overline{(s1)} = \underline{(sx)})n ⇒ \overline{(s1)} = \underline{(sy)}; ∀\underline{(sx)}: ∀\underline{(sx1)}: ∀\underline{(sy1)}: \underline{(sx)} ∈ \underline{(sx1)} ⊢ \overline{(s1)} = \underline{(sx)} ⊢ SameMember(2) ⊳ \overline{(s1)} = \underline{(sx)} ⊳ \underline{(sx)} ∈ \underline{(sx1)} ≫ \overline{(s1)} ∈ \underline{(sx1)}; ToBinaryUnion(1) ⊳ \overline{(s1)} ∈ \underline{(sx1)} ≫ \overline{(s1)} ∈ Union(\{\underline{(sx1)}, \underline{(sy1)}\}); ∀\underline{(sx1)}: ∀\underline{(sy)}: ∀\underline{(sy1)}: \underline{(sy)} ∈ \underline{(sy1)} ⊢ \overline{(s1)} = \underline{(sy)} ⊢ SameMember(2) ⊳ \overline{(s1)} = \underline{(sy)} ⊳ \underline{(sy)} ∈ \underline{(sy1)} ≫ \overline{(s1)} ∈ \underline{(sy1)}; ToBinaryUnion(2) ⊳ \overline{(s1)} ∈ \underline{(sy1)} ≫ \overline{(s1)} ∈ Union(\{\underline{(sx1)}, \underline{(sy1)}\}); Ded ⊳ ∀\underline{(sx)}: ∀\underline{(sx1)}: ∀\underline{(sy1)}: \underline{(sx)} ∈ \underline{(sx1)} ⊢ \overline{(s1)} = \underline{(sx)} ⊢ \overline{(s1)} ∈ Union(\{\underline{(sx1)}, \underline{(sy1)}\}) ≫ \underline{(sx)} ∈ \underline{(sx1)} ⇒ \overline{(s1)} = \underline{(sx)} ⇒ \overline{(s1)} ∈ Union(\{\underline{(sx1)}, \underline{(sy1)}\}); MP ⊳ \underline{(sx)} ∈ \underline{(sx1)} ⇒ \overline{(s1)} = \underline{(sx)} ⇒ \overline{(s1)} ∈ Union(\{\underline{(sx1)}, \underline{(sy1)}\}) ⊳ \underline{(sx)} ∈ \underline{(sx1)} ≫ \overline{(s1)} = \underline{(sx)} ⇒ \overline{(s1)} ∈ Union(\{\underline{(sx1)}, \underline{(sy1)}\}); Ded ⊳ ∀\underline{(sx1)}: ∀\underline{(sy)}: ∀\underline{(sy1)}: \underline{(sy)} ∈ \underline{(sy1)} ⊢ \overline{(s1)} = \underline{(sy)} ⊢ \overline{(s1)} ∈ Union(\{\underline{(sx1)}, \underline{(sy1)}\}) ≫ \underline{(sy)} ∈ \underline{(sy1)} ⇒ \overline{(s1)} = \underline{(sy)} ⇒ \overline{(s1)} ∈ Union(\{\underline{(sx1)}, \underline{(sy1)}\}); MP ⊳ \underline{(sy)} ∈ \underline{(sy1)} ⇒ \overline{(s1)} = \underline{(sy)} ⇒ \overline{(s1)} ∈ Union(\{\underline{(sx1)}, \underline{(sy1)}\}) ⊳ \underline{(sy)} ∈ \underline{(sy1)} ≫ \overline{(s1)} = \underline{(sy)} ⇒ \overline{(s1)} ∈ Union(\{\underline{(sx1)}, \underline{(sy1)}\}); FromDisjuncts ⊳ ˙¬ (\overline{(s1)} = \underline{(sx)})n ⇒ \overline{(s1)} = \underline{(sy)} ⊳ \overline{(s1)} = \underline{(sx)} ⇒ \overline{(s1)} ∈ Union(\{\underline{(sx1)}, \underline{(sy1)}\}) ⊳ \overline{(s1)} = \underline{(sy)} ⇒ \overline{(s1)} ∈ Union(\{\underline{(sx1)}, \underline{(sy1)}\}) ≫ \overline{(s1)} ∈ Union(\{\underline{(sx1)}, \underline{(sy1)}\}); ∀\underline{(sx)}: ∀\underline{(sx1)}: ∀\underline{(sy)}: ∀\underline{(sy1)}: ∀\underline{(sz)}: Ded ⊳ ∀\underline{(sx)}: ∀\underline{(sx1)}: ∀\underline{(sy)}: ∀\underline{(sy1)}: ∀\underline{(sz)}: \underline{(sx)} ∈ \underline{(sx1)} ⊢ \underline{(sy)} ∈ \underline{(sy1)} ⊢ \underline{(sz)} ∈ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ⊢ \overline{(s1)} ∈ \underline{(sz)} ⊢ \overline{(s1)} ∈ Union(\{\underline{(sx1)}, \underline{(sy1)}\}) ≫ \underline{(sx)} ∈ \underline{(sx1)} ⇒ \underline{(sy)} ∈ \underline{(sy1)} ⇒ \underline{(sz)} ∈ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ⇒ \overline{(s1)} ∈ \underline{(sz)} ⇒ \overline{(s1)} ∈ Union(\{\underline{(sx1)}, \underline{(sy1)}\}); \underline{(sx)} ∈ \underline{(sx1)} ⊢ \underline{(sy)} ∈ \underline{(sy1)} ⊢ \underline{(sz)} ∈ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ⊢ MP3 ⊳ \underline{(sx)} ∈ \underline{(sx1)} ⇒ \underline{(sy)} ∈ \underline{(sy1)} ⇒ \underline{(sz)} ∈ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ⇒ \overline{(s1)} ∈ \underline{(sz)} ⇒ \overline{(s1)} ∈ Union(\{\underline{(sx1)}, \underline{(sy1)}\})⊳\underline{(sx)} ∈ \underline{(sx1)} ⊳ \underline{(sy)} ∈ \underline{(sy1)} ⊳ \underline{(sz)} ∈ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ≫ \overline{(s1)} ∈ \underline{(sz)} ⇒ \overline{(s1)} ∈ Union(\{\underline{(sx1)}, \underline{(sy1)}\}); Gen ⊳ \overline{(s1)} ∈ \underline{(sz)} ⇒ \overline{(s1)} ∈ Union(\{\underline{(sx1)}, \underline{(sy1)}\}) ≫ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \underline{(sz)} ⇒ \overline{(s1)} ∈ Union(\{\underline{(sx1)}, \underline{(sy1)}\}); Repetition⊳∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \underline{(sz)} ⇒ \overline{(s1)} ∈ Union(\{\underline{(sx1)}, \underline{(sy1)}\}) ≫ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \underline{(sz)} ⇒ \overline{(s1)} ∈ Union(\{\underline{(sx1)}, \underline{(sy1)}\})⌉, p_{0}, c)]

[ToCartProd(Helper) →^{stmt} SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sx1)}: ∀\underline{(sy)}: ∀\underline{(sy1)}: ∀\underline{(sz)}: \underline{(sx)} ∈ \underline{(sx1)} ⊢ \underline{(sy)} ∈ \underline{(sy1)} ⊢ \underline{(sz)} ∈ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ⊢ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \underline{(sz)} ⇒ \overline{(s1)} ∈ Union(\{\underline{(sx1)}, \underline{(sy1)}\})]

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

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

ToCartProd

[ToCartProd →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sx1)}: ∀\underline{(sy)}: ∀\underline{(sy1)}: \underline{(sx)} ∈ \underline{(sx1)} ⊢ \underline{(sy)} ∈ \underline{(sy1)} ⊢ \overline{(s1)} ∈ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ⊢ ToCartProd(Helper) ⊳ \underline{(sx)} ∈ \underline{(sx1)} ⊳ \underline{(sy)} ∈ \underline{(sy1)} ⊳ \overline{(s1)} ∈ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ≫ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \overline{(s1)} ⇒ \overline{(s1)} ∈ Union(\{\underline{(sx1)}, \underline{(sy1)}\}); Formula2Power ⊳ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \overline{(s1)} ⇒ \overline{(s1)} ∈ Union(\{\underline{(sx1)}, \underline{(sy1)}\}) ≫ \overline{(s1)} ∈ P(Union(\{\underline{(sx1)}, \underline{(sy1)}\})); ∀\underline{(sx)}: ∀\underline{(sx1)}: ∀\underline{(sy)}: ∀\underline{(sy1)}: Ded ⊳ ∀\underline{(sx)}: ∀\underline{(sx1)}: ∀\underline{(sy)}: ∀\underline{(sy1)}: \underline{(sx)} ∈ \underline{(sx1)} ⊢ \underline{(sy)} ∈ \underline{(sy1)} ⊢ \overline{(s1)} ∈ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ⊢ \overline{(s1)} ∈ P(Union(\{\underline{(sx1)}, \underline{(sy1)}\})) ≫ \underline{(sx)} ∈ \underline{(sx1)} ⇒ \underline{(sy)} ∈ \underline{(sy1)} ⇒ \overline{(s1)} ∈ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ⇒ \overline{(s1)} ∈ P(Union(\{\underline{(sx1)}, \underline{(sy1)}\})); \underline{(sx)} ∈ \underline{(sx1)} ⊢ \underline{(sy)} ∈ \underline{(sy1)} ⊢ MP2 ⊳ \underline{(sx)} ∈ \underline{(sx1)} ⇒ \underline{(sy)} ∈ \underline{(sy1)} ⇒ \overline{(s1)} ∈ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ⇒ \overline{(s1)} ∈ P(Union(\{\underline{(sx1)}, \underline{(sy1)}\})) ⊳ \underline{(sx)} ∈ \underline{(sx1)} ⊳ \underline{(sy)} ∈ \underline{(sy1)} ≫ \overline{(s1)} ∈ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ⇒ \overline{(s1)} ∈ P(Union(\{\underline{(sx1)}, \underline{(sy1)}\})); Gen ⊳ \overline{(s1)} ∈ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ⇒ \overline{(s1)} ∈ P(Union(\{\underline{(sx1)}, \underline{(sy1)}\})) ≫ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ⇒ \overline{(s1)} ∈ P(Union(\{\underline{(sx1)}, \underline{(sy1)}\})); Repetition ⊳ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ⇒ \overline{(s1)} ∈ P(Union(\{\underline{(sx1)}, \underline{(sy1)}\})) ≫ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ⇒ \overline{(s1)} ∈ P(Union(\{\underline{(sx1)}, \underline{(sy1)}\})); Formula2Power ⊳ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ⇒ \overline{(s1)} ∈ P(Union(\{\underline{(sx1)}, \underline{(sy1)}\})) ≫ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ∈ P(P(Union(\{\underline{(sx1)}, \underline{(sy1)}\}))); eqReflexivity ≫ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\}; JoinConjuncts ⊳ \underline{(sx)} ∈ \underline{(sx1)} ⊳ \underline{(sy)} ∈ \underline{(sy1)} ≫ ˙¬ (\underline{(sx)} ∈ \underline{(sx1)} ⇒ ˙¬ (\underline{(sy)} ∈ \underline{(sy1)})n)n; JoinConjuncts ⊳ ˙¬ (\underline{(sx)} ∈ \underline{(sx1)} ⇒ ˙¬ (\underline{(sy)} ∈ \underline{(sy1)})n)n ⊳ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ≫ ˙¬ ( ˙¬ (\underline{(sx)} ∈ \underline{(sx1)} ⇒ ˙¬ (\underline{(sy)} ∈ \underline{(sy1)})n)n ⇒ ˙¬ (\{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\})n)n; IntroExist @ \underline{(sy)} ⊳ ˙¬ ( ˙¬ (\underline{(sx)} ∈ \underline{(sx1)} ⇒ ˙¬ (\underline{(sy)} ∈ \underline{(sy1)})n)n ⇒ ˙¬ (\{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\})n)n ≫ ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\underline{(sx)} ∈ \underline{(sx1)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy1)})n)n ⇒ ˙¬ (\{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \overline{(op2)}\}\})n)n)n)n; IntroExist @ \underline{(sx)} ⊳ ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\underline{(sx)} ∈ \underline{(sx1)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy1)})n)n ⇒ ˙¬ (\{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \overline{(op2)}\}\})n)n)n)n ≫ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx1)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy1)})n)n ⇒ ˙¬ (\{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n; Repetition ⊳ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx1)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy1)})n)n ⇒ ˙¬ (\{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ≫ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx1)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy1)})n)n ⇒ ˙¬ (\{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n; Formula2Sep ⊳ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ∈ P(P(Union(\{\underline{(sx1)}, \underline{(sy1)}\}))) ⊳ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx1)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy1)})n)n ⇒ ˙¬ (\{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ≫ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ∈ \{ph ∈ P(P(Union(\{\underline{(sx1)}, \underline{(sy1)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx1)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy1)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}; Repetition ⊳ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ∈ \{ph ∈ P(P(Union(\{\underline{(sx1)}, \underline{(sy1)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx1)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy1)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ≫ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ∈ \{ph ∈ P(P(Union(\{\underline{(sx1)}, \underline{(sy1)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx1)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy1)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}⌉, p_{0}, c)] [ToCartProd →^{stmt} SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sx1)}: ∀\underline{(sy)}: ∀\underline{(sy1)}: \underline{(sx)} ∈ \underline{(sx1)} ⊢ \underline{(sy)} ∈ \underline{(sy1)} ⊢ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ∈ \{ph ∈ P(P(Union(\{\underline{(sx1)}, \underline{(sy1)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx1)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy1)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}] [ToCartProd →^{tex} “ToCartProd”]

[ToCartProd →^{pyk} “lemma toCartProd”]

NonreciprocalToRight(Eq)

[NonreciprocalToRight(Eq) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ (\underline{y} = 0)n ⊢ (\underline{x} ∗ \underline{y}) = \underline{z} ⊢ eqMultiplication ⊳ (\underline{x} ∗ \underline{y}) = \underline{z} ≫ ((\underline{x} ∗ \underline{y}) ∗ rec\underline{y}) = (\underline{z} ∗ rec\underline{y}); x = x ∗ y ∗ (1/y) ⊳ ˙¬ (\underline{y} = 0)n ≫ \underline{x} = ((\underline{x} ∗ \underline{y}) ∗ rec\underline{y}); eqTransitivity ⊳ \underline{x} = ((\underline{x} ∗ \underline{y}) ∗ rec\underline{y}) ⊳ ((\underline{x} ∗ \underline{y}) ∗ rec\underline{y}) = (\underline{z} ∗ rec\underline{y}) ≫ \underline{x} = (\underline{z} ∗ rec\underline{y})⌉, p_{0}, c)]

[NonreciprocalToRight(Eq) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: ˙¬ (\underline{y} = 0)n ⊢ (\underline{x} ∗ \underline{y}) = \underline{z} ⊢ \underline{x} = (\underline{z} ∗ rec\underline{y})]

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

[NonreciprocalToRight(Eq) →^{pyk} “lemma nonreciprocalToRight(Eq)”]

NonreciprocalToLeft(Eq)(1term)

[NonreciprocalToLeft(Eq)(1term) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: 1 = (\underline{x} ∗ \underline{y}) ⊢ eqSymmetry ⊳ 1 = (\underline{x} ∗ \underline{y}) ≫ (\underline{x} ∗ \underline{y}) = 1; NonreciprocalToRight(Eq)(1term) ⊳ (\underline{x} ∗ \underline{y}) = 1 ≫ \underline{x} = rec\underline{y}; eqSymmetry ⊳ \underline{x} = rec\underline{y} ≫ rec\underline{y} = \underline{x}⌉, p_{0}, c)]

[NonreciprocalToLeft(Eq)(1term) →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: 1 = (\underline{x} ∗ \underline{y}) ⊢ rec\underline{y} = \underline{x}]

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

[NonreciprocalToLeft(Eq)(1term) →^{pyk} “lemma nonreciprocalToLeft(Eq)(1 term)”]

SameReciprocal

[SameReciprocal →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ (\underline{x} = 0)n ⊢ \underline{x} = \underline{y} ⊢ times1Left ≫ (1 ∗ \underline{x}) = \underline{x}; eqTransitivity ⊳ (1 ∗ \underline{x}) = \underline{x} ⊳ \underline{x} = \underline{y} ≫ (1 ∗ \underline{x}) = \underline{y}; NonreciprocalToRight(Eq) ⊳ ˙¬ (\underline{x} = 0)n ⊳ (1 ∗ \underline{x}) = \underline{y} ≫ 1 = (\underline{y} ∗ rec\underline{x}); timesCommutativity ≫ (\underline{y} ∗ rec\underline{x}) = (rec\underline{x} ∗ \underline{y}); eqTransitivity ⊳ 1 = (\underline{y} ∗ rec\underline{x}) ⊳ (\underline{y} ∗ rec\underline{x}) = (rec\underline{x} ∗ \underline{y}) ≫ 1 = (rec\underline{x} ∗ \underline{y}); NonreciprocalToLeft(Eq)(1term) ⊳ 1 = (rec\underline{x} ∗ \underline{y}) ≫ rec\underline{y} = rec\underline{x}; eqSymmetry ⊳ rec\underline{y} = rec\underline{x} ≫ rec\underline{x} = rec\underline{y}⌉, p_{0}, c)]

[SameReciprocal →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ˙¬ (\underline{x} = 0)n ⊢ \underline{x} = \underline{y} ⊢ rec\underline{x} = rec\underline{y}]

[SameReciprocal →^{tex} “SameReciprocal”]

[SameReciprocal →^{pyk} “lemma sameReciprocal”]

CPseparationIsRelation

[CPseparationIsRelation →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{a}: ∀\underline{(sx)}: ∀\underline{(sy)}: \overline{(s1)} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{\underline{(sx)}, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | \underline{a}\} ⊢ Separation2formula(1) ⊳ \overline{(s1)} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{\underline{(sx)}, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | \underline{a}\} ≫ \overline{(s1)} ∈ \{ph ∈ P(P(Union(\{\underline{(sx)}, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}; ∀\underline{a}: ∀\underline{(sx)}: ∀\underline{(sy)}: Ded ⊳ ∀\underline{a}: ∀\underline{(sx)}: ∀\underline{(sy)}: \overline{(s1)} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{\underline{(sx)}, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | \underline{a}\} ⊢ \overline{(s1)} ∈ \{ph ∈ P(P(Union(\{\underline{(sx)}, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ≫ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{\underline{(sx)}, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | \underline{a}\} ⇒ \overline{(s1)} ∈ \{ph ∈ P(P(Union(\{\underline{(sx)}, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}; Repetition ⊳ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{\underline{(sx)}, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | \underline{a}\} ⇒ \overline{(s1)} ∈ \{ph ∈ P(P(Union(\{\underline{(sx)}, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ≫ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{\underline{(sx)}, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | \underline{a}\} ⇒ \overline{(s1)} ∈ \{ph ∈ P(P(Union(\{\underline{(sx)}, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}; CartProdIsRelation ≫ ∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \{ph ∈ P(P(Union(\{\underline{(sx)}, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n; SubsetIsRelation ⊳ ∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \{ph ∈ P(P(Union(\{\underline{(sx)}, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⊳ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{\underline{(sx)}, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | \underline{a}\} ⇒ \overline{(s1)} ∈ \{ph ∈ P(P(Union(\{\underline{(sx)}, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ≫ ∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{\underline{(sx)}, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | \underline{a}\} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n⌉, p_{0}, c)]

[CPseparationIsRelation →^{stmt} SystemQ ⊢ ∀\underline{a}: ∀\underline{(sx)}: ∀\underline{(sy)}: ∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{\underline{(sx)}, \underline{(sy)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | \underline{a}\} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n]

[CPseparationIsRelation →^{tex} “CPseparationIsRelation”]

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

OrderedPairEquality

[OrderedPairEquality →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sx1)}: ∀\underline{(sy)}: ∀\underline{(sy1)}: ∀\underline{(sz)}: ∀\underline{(sz1)}: ∀\underline{(su)}: ∀\underline{(su1)}: \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sx1)}\}\} = \{\{\underline{(sy)}, \underline{(sy)}\}, \{\underline{(sy)}, \underline{(sy1)}\}\} ⊢ \{\{\underline{(sz)}, \underline{(sz)}\}, \{\underline{(sz)}, \underline{(sz1)}\}\} = \{\{\underline{(su)}, \underline{(su)}\}, \{\underline{(su)}, \underline{(su1)}\}\} ⊢ \underline{(sx)} = \underline{(sz)} ⊢ FromOrderedPair(1) ⊳ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sx1)}\}\} = \{\{\underline{(sy)}, \underline{(sy)}\}, \{\underline{(sy)}, \underline{(sy1)}\}\} ≫ \underline{(sx)} = \underline{(sy)}; eqSymmetry ⊳ \underline{(sx)} = \underline{(sy)} ≫ \underline{(sy)} = \underline{(sx)}; FromOrderedPair(1) ⊳ \{\{\underline{(sz)}, \underline{(sz)}\}, \{\underline{(sz)}, \underline{(sz1)}\}\} = \{\{\underline{(su)}, \underline{(su)}\}, \{\underline{(su)}, \underline{(su1)}\}\} ≫ \underline{(sz)} = \underline{(su)}; eqTransitivity4 ⊳ \underline{(sy)} = \underline{(sx)} ⊳ \underline{(sx)} = \underline{(sz)} ⊳ \underline{(sz)} = \underline{(su)} ≫ \underline{(sy)} = \underline{(su)}⌉, p_{0}, c)]

[OrderedPairEquality →^{stmt} SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sx1)}: ∀\underline{(sy)}: ∀\underline{(sy1)}: ∀\underline{(sz)}: ∀\underline{(sz1)}: ∀\underline{(su)}: ∀\underline{(su1)}: \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sx1)}\}\} = \{\{\underline{(sy)}, \underline{(sy)}\}, \{\underline{(sy)}, \underline{(sy1)}\}\} ⊢ \{\{\underline{(sz)}, \underline{(sz)}\}, \{\underline{(sz)}, \underline{(sz1)}\}\} = \{\{\underline{(su)}, \underline{(su)}\}, \{\underline{(su)}, \underline{(su1)}\}\} ⊢ \underline{(sx)} = \underline{(sz)} ⊢ \underline{(sy)} = \underline{(su)}] [OrderedPairEquality →^{tex} “OrderedPairEquality”]

[OrderedPairEquality →^{pyk} “lemma orderedPairEquality”]

ReciprocalIsFunction

[ReciprocalIsFunction →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{m}: ∀\underline{(fx)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\} ⊢ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\} ⊢ \overline{(f1)} = \overline{(f3)} ⊢ Repetition ⊳ \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\} ≫ \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}; Separation2formula(2) ⊳ \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ {ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\} ≫ ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n; Repetition ⊳ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\} ≫ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}; Separation2formula(2) ⊳ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ {ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\} ≫ ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n; ∀\underline{m}: ∀\underline{m}: ∀\underline{(fx)}: \overline{(f1)} = \overline{(f3)} ⊢ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n ⊢ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n ⊢ FirstConjunct ⊳ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n ≫ ˙¬ (\underline{(fx)}[\underline{m}] = 0)n; SecondConjunct ⊳ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n ≫ \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\}; SecondConjunct ⊳ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n ≫ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\}; FromOrderedPair(1) ⊳ \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\} ≫ \overline{(f1)} = \underline{m}; eqSymmetry ⊳ \overline{(f1)} = \underline{m} ≫ \underline{m} = \overline{(f1)}; FromOrderedPair(1) ⊳ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\} ≫ \overline{(f3)} = \underline{m}; eqTransitivity4 ⊳ \underline{m} = \overline{(f1)} ⊳ \overline{(f1)} = \overline{(f3)} ⊳ \overline{(f3)} = \underline{m} ≫ \underline{m} = \underline{m}; SameSeries ⊳ \underline{m} = \underline{m} ≫ \underline{(fx)}[\underline{m}] = \underline{(fx)}[\underline{m}]; SameReciprocal ⊳ ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⊳ \underline{(fx)}[\underline{m}] = \underline{(fx)}[\underline{m}] ≫ rec\underline{(fx)}[\underline{m}] = rec\underline{(fx)}[\underline{m}]; FromOrderedPair(2) ⊳ \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\} ≫ \overline{(f2)} = rec\underline{(fx)}[\underline{m}]; FromOrderedPair(2) ⊳ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\} ≫ \overline{(f4)} = rec\underline{(fx)}[\underline{m}]; eqSymmetry ⊳ \overline{(f4)} = rec\underline{(fx)}[\underline{m}] ≫ rec\underline{(fx)}[\underline{m}] = \overline{(f4)}; eqTransitivity4 ⊳ \overline{(f2)} = rec\underline{(fx)}[\underline{m}] ⊳ rec\underline{(fx)}[\underline{m}] = rec\underline{(fx)}[\underline{m}] ⊳ rec\underline{(fx)}[\underline{m}] = \overline{(f4)} ≫ \overline{(f2)} = \overline{(f4)}; ∀\underline{m}: ∀\underline{m}: ∀\underline{(fx)}: \overline{(f1)} = \overline{(f3)} ⊢ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n ⊢ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n ⊢ FirstConjunct ⊳ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n ≫ ˙¬ (\underline{(fx)}[\underline{m}] = 0)n; SecondConjunct ⊳ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n ≫ \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\}; FirstConjunct ⊳ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n ≫ \underline{(fx)}[\underline{m}] = 0; SecondConjunct ⊳ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n ≫ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\}; OrderedPairEquality ⊳ \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\} ⊳ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\} ⊳ \overline{(f1)} = \overline{(f3)} ≫ \underline{m} = \underline{m}; SameSeries ⊳ \underline{m} = \underline{m} ≫ \underline{(fx)}[\underline{m}] = \underline{(fx)}[\underline{m}]; eqTransitivity ⊳ \underline{(fx)}[\underline{m}] = \underline{(fx)}[\underline{m}] ⊳ \underline{(fx)}[\underline{m}] = 0 ≫ \underline{(fx)}[\underline{m}] = 0; FromContradiction ⊳ \underline{(fx)}[\underline{m}] = 0 ⊳ ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ≫ \overline{(f2)} = \overline{(f4)}; ∀\underline{m}: ∀\underline{m}: ∀\underline{(fx)}: \overline{(f1)} = \overline{(f3)} ⊢ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n ⊢ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n ⊢ FirstConjunct ⊳ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n ≫ \underline{(fx)}[\underline{m}] = 0; SecondConjunct ⊳ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n ≫ \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\}; FirstConjunct ⊳ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n ≫ ˙¬ (\underline{(fx)}[\underline{m}] = 0)n; SecondConjunct ⊳ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n ≫ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\}; OrderedPairEquality ⊳ \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\} ⊳ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\} ⊳ \overline{(f1)} = \overline{(f3)} ≫ \underline{m} = \underline{m}; SameSeries ⊳ \underline{m} = \underline{m} ≫ \underline{(fx)}[\underline{m}] = \underline{(fx)}[\underline{m}]; eqSymmetry ⊳ \underline{(fx)}[\underline{m}] = \underline{(fx)}[\underline{m}] ≫ \underline{(fx)}[\underline{m}] = \underline{(fx)}[\underline{m}]; eqTransitivity ⊳ \underline{(fx)}[\underline{m}] = \underline{(fx)}[\underline{m}] ⊳ \underline{(fx)}[\underline{m}] = 0 ≫ \underline{(fx)}[\underline{m}] = 0; FromContradiction ⊳ \underline{(fx)}[\underline{m}] = 0 ⊳ ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ≫ \overline{(f2)} = \overline{(f4)}; ∀\underline{m}: ∀\underline{m}: ∀\underline{(fx)}: ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n ⊢ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n ⊢ SecondConjunct ⊳ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n ≫ \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\}; SecondConjunct ⊳ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n ≫ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\}; FromOrderedPair(2) ⊳ \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\} ≫ \overline{(f2)} = 0; FromOrderedPair(2) ⊳ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\} ≫ \overline{(f4)} = 0; eqSymmetry ⊳ \overline{(f4)} = 0 ≫ 0 = \overline{(f4)}; eqTransitivity ⊳ \overline{(f2)} = 0 ⊳ 0 = \overline{(f4)} ≫ \overline{(f2)} = \overline{(f4)}; ∀\underline{m}: ∀\underline{m}: ∀\underline{(fx)}: \overline{(f1)} = \overline{(f3)} ⊢ ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n ⊢ ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n ⊢ Ded ⊳ ∀\underline{m}: ∀\underline{m}: ∀\underline{(fx)}: \overline{(f1)} = \overline{(f3)} ⊢ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n ⊢ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n ⊢ \overline{(f2)} = \overline{(f4)} ≫ \overline{(f1)} = \overline{(f3)} ⇒ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n ⇒ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n ⇒ \overline{(f2)} = \overline{(f4)}; MP ⊳ \overline{(f1)} = \overline{(f3)} ⇒ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n ⇒ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n ⇒ \overline{(f2)} = \overline{(f4)} ⊳ \overline{(f1)} = \overline{(f3)} ≫ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n ⇒ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n ⇒ \overline{(f2)} = \overline{(f4)}; Ded ⊳ ∀\underline{m}: ∀\underline{m}: ∀\underline{(fx)}: \overline{(f1)} = \overline{(f3)} ⊢ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n ⊢ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n ⊢ \overline{(f2)} = \overline{(f4)} ≫ \overline{(f1)} = \overline{(f3)} ⇒ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n ⇒ \overline{(f2)} = \overline{(f4)}; MP ⊳ \overline{(f1)} = \overline{(f3)} ⇒ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n ⇒ \overline{(f2)} = \overline{(f4)} ⊳ \overline{(f1)} = \overline{(f3)} ≫ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n ⇒ \overline{(f2)} = \overline{(f4)}; Ded ⊳ ∀\underline{m}: ∀\underline{m}: ∀\underline{(fx)}: \overline{(f1)} = \overline{(f3)} ⊢ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n ⊢ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n ⊢ \overline{(f2)} = \overline{(f4)} ≫ \overline{(f1)} = \overline{(f3)} ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n ⇒ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n ⇒ \overline{(f2)} = \overline{(f4)}; MP ⊳ \overline{(f1)} = \overline{(f3)} ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n ⇒ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n ⇒ \overline{(f2)} = \overline{(f4)} ⊳ \overline{(f1)} = \overline{(f3)} ≫ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n ⇒ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n ⇒ \overline{(f2)} = \overline{(f4)}; Ded ⊳ ∀\underline{m}: ∀\underline{m}: ∀\underline{(fx)}: ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n ⊢ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n ⊢ \overline{(f2)} = \overline{(f4)} ≫ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n ⇒ \overline{(f2)} = \overline{(f4)}; From2 ∗ 2Disjuncts ⊳ ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n ⊳ ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n ⊳ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n ⇒ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n ⇒ \overline{(f2)} = \overline{(f4)} ⊳ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n ⇒ \overline{(f2)} = \overline{(f4)}⊳ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n ⇒ ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n ⇒ \overline{(f2)} = \overline{(f4)} ⊳ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n ⇒ \overline{(f2)} = \overline{(f4)} ≫ \overline{(f2)} = \overline{(f4)}; Ded ⊳ ∀\underline{m}: ∀\underline{m}: ∀\underline{(fx)}: \overline{(f1)} = \overline{(f3)} ⊢ ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n ⊢ ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n ⊢ \overline{(f2)} = \overline{(f4)} ≫ \overline{(f1)} = \overline{(f3)} ⇒ ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n ⇒ ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n ⇒ \overline{(f2)} = \overline{(f4)}; MP ⊳ \overline{(f1)} = \overline{(f3)} ⇒ ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n ⇒ ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n ⇒ \overline{(f2)} = \overline{(f4)} ⊳ \overline{(f1)} = \overline{(f3)} ≫ ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n ⇒ ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n ⇒ \overline{(f2)} = \overline{(f4)}; ExistMP2 ⊳ ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n ⇒ ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n ⇒ \overline{(f2)} = \overline{(f4)} ⊳ ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n ⊳ ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n ≫ \overline{(f2)} = \overline{(f4)}; ∀\underline{m}: ∀\underline{m}: ∀\underline{(fx)}: Ded ⊳ ∀\underline{m}: ∀\underline{m}: ∀\underline{(fx)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\} ⊢ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\} ⊢ \overline{(f1)} = \overline{(f3)} ⊢ \overline{(f2)} = \overline{(f4)} ≫ ∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)}⌉, p_{0}, c)]

[ReciprocalIsFunction →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{m}: ∀\underline{(fx)}: ∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ {ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)}] [ReciprocalIsFunction →^{tex} “ReciprocalIsFunction”]

[ReciprocalIsFunction →^{pyk} “lemma reciprocalIsFunction”]

ReciprocalIsTotal

[ReciprocalIsTotal →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: \underline{(fx)}[\overline{(s1)}] = 0 ⊢ \overline{(s1)} ∈ N ⊢ RationalType ≫ 0 ∈ Q; ToCartProd ⊳ \overline{(s1)} ∈ N ⊳ 0 ∈ Q ≫ \{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, 0\}\} ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}; eqReflexivity ≫ \{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, 0\}\} = \{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, 0\}\}; JoinConjuncts ⊳ \underline{(fx)}[\overline{(s1)}] = 0 ⊳ \{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, 0\}\} = \{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, 0\}\} ≫ ˙¬ (\underline{(fx)}[\overline{(s1)}] = 0 ⇒ ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, 0\}\} = \{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, 0\}\})n)n; WeakenOr1 ⊳ ˙¬ (\underline{(fx)}[\overline{(s1)}] = 0 ⇒ ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, 0\}\} = \{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, 0\}\})n)n ≫ ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\overline{(s1)}] = 0)n ⇒ ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, 0\}\} = \{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, rec\underline{(fx)}[\overline{(s1)}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\overline{(s1)}] = 0 ⇒ ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, 0\}\} = \{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, 0\}\})n)n; IntroExist @ \overline{(s1)} ⊳ ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\overline{(s1)}] = 0)n ⇒ ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, 0\}\} = \{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, rec\underline{(fx)}[\overline{(s1)}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\overline{(s1)}] = 0 ⇒ ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, 0\}\} = \{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, 0\}\})n)n ≫ ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, 0\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, 0\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n; Formula2Sep ⊳ \{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, 0\}\} ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ⊳ ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, 0\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, 0\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n ≫ \{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, 0\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}; IntroExist @ 0 ⊳ \{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, 0\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\} ≫ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\})n)n; ∀\underline{(fx)}: ˙¬ (\underline{(fx)}[\overline{(s1)}] = 0)n ⊢ \overline{(s1)} ∈ N ⊢ SeriesType ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n; ValueType ⊳ \overline{(s1)} ∈ N ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ≫ \underline{(fx)}[\overline{(s1)}] ∈ Q; QisClosed(Reciprocal) ⊳ ˙¬ (\underline{(fx)}[\overline{(s1)}] = 0)n⊳\underline{(fx)}[\overline{(s1)}] ∈ Q ≫ rec\underline{(fx)}[\overline{(s1)}] ∈ Q; ToCartProd⊳\overline{(s1)} ∈ N⊳rec\underline{(fx)}[\overline{(s1)}] ∈ Q ≫ \{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, rec\underline{(fx)}[\overline{(s1)}]\}\} ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}; eqReflexivity ≫ \{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, rec\underline{(fx)}[\overline{(s1)}]\}\} = \{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, rec\underline{(fx)}[\overline{(s1)}]\}\}; JoinConjuncts ⊳ ˙¬ (\underline{(fx)}[\overline{(s1)}] = 0)n ⊳ \{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, rec\underline{(fx)}[\overline{(s1)}]\}\} = \{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, rec\underline{(fx)}[\overline{(s1)}]\}\} ≫ ˙¬ ( ˙¬ (\underline{(fx)}[\overline{(s1)}] = 0)n ⇒ ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, rec\underline{(fx)}[\overline{(s1)}]\}\} = \{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, rec\underline{(fx)}[\overline{(s1)}]\}\})n)n; WeakenOr2 ⊳ ˙¬ ( ˙¬ (\underline{(fx)}[\overline{(s1)}] = 0)n ⇒ ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, rec\underline{(fx)}[\overline{(s1)}]\}\} = \{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, rec\underline{(fx)}[\overline{(s1)}]\}\})n)n ≫ ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\overline{(s1)}] = 0)n ⇒ ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, rec\underline{(fx)}[\overline{(s1)}]\}\} = \{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, rec\underline{(fx)}[\overline{(s1)}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\overline{(s1)}] = 0 ⇒ ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, rec\underline{(fx)}[\overline{(s1)}]\}\} = \{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, 0\}\})n)n; IntroExist @ \overline{(s1)} ⊳ ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\overline{(s1)}] = 0)n ⇒ ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, rec\underline{(fx)}[\overline{(s1)}]\}\} = \{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, rec\underline{(fx)}[\overline{(s1)}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\overline{(s1)}] = 0 ⇒ ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, rec\underline{(fx)}[\overline{(s1)}]\}\} = \{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, 0\}\})n)n ≫ ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, rec\underline{(fx)}[\overline{(s1)}]\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, rec\underline{(fx)}[\overline{(s1)}]\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n; Formula2Sep ⊳ \{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, rec\underline{(fx)}[\overline{(s1)}]\}\} ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ⊳ ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, rec\underline{(fx)}[\overline{(s1)}]\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, rec\underline{(fx)}[\overline{(s1)}]\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n ≫ \{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, rec\underline{(fx)}[\overline{(s1)}]\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\}; IntroExist @ rec\underline{(fx)}[\overline{(s1)}] ⊳ \{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, rec\underline{(fx)}[\overline{(s1)}]\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\} ≫ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\})n)n; ∀\underline{(fx)}: Ded ⊳ ∀\underline{(fx)}: \underline{(fx)}[\overline{(s1)}] = 0 ⊢ \overline{(s1)} ∈ N ⊢ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\})n)n ≫ \underline{(fx)}[\overline{(s1)}] = 0 ⇒ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\})n)n; Ded ⊳ ∀\underline{(fx)}: ˙¬ (\underline{(fx)}[\overline{(s1)}] = 0)n ⊢ \overline{(s1)} ∈ N ⊢ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\})n)n ≫ ˙¬ (\underline{(fx)}[\overline{(s1)}] = 0)n ⇒ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\})n)n; FromNegations ⊳ \underline{(fx)}[\overline{(s1)}] = 0 ⇒ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\})n)n ⊳ ˙¬ (\underline{(fx)}[\overline{(s1)}] = 0)n ⇒ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\})n)n ≫ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\})n)n⌉, p_{0}, c)]

[ReciprocalIsTotal →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\})n)n]

[ReciprocalIsTotal →^{tex} “ReciprocalIsTotal”] [ReciprocalIsTotal →^{pyk} “lemma reciprocalIsTotal”]

ReciprocalIsRationalSeries

[ReciprocalIsRationalSeries →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: CPseparationIsRelation ≫ ∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n; ReciprocalIsFunction ≫ ∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)}; ReciprocalIsTotal ≫ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\})n)n; ToSeries ⊳ ∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⊳ ∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)} ⊳ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\})n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\underline{m}: ˙¬ ( ˙¬ ( ˙¬ ( ˙¬ (\underline{(fx)}[\underline{m}] = 0)n ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, rec\underline{(fx)}[\underline{m}]\}\})n)n)n ⇒ ˙¬ (\underline{(fx)}[\underline{m}] = 0 ⇒ ˙¬ (f_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, 0\}\})n)n)n)n\})n)n)n)n⌉, p_{0}, c)]

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

[ReciprocalIsRationalSeries →^{tex} “ReciprocalIsRationalSeries”]

[ReciprocalIsRationalSeries →^{pyk} “lemma reciprocalIsRationalSeries”]

CrsIsRelation

[CrsIsRelation →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: \overline{(s1)} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⊢ Repetition ⊳ \overline{(s1)} ∈ {ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ≫ \overline{(s1)} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\}; Sep2Formula ⊳ \overline{(s1)} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ≫ ˙¬ (\overline{(s1)} ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (\overline{(s1)} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n)n)n; FirstConjunct ⊳ ˙¬ (\overline{(s1)} ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (\overline{(s1)} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n)n)n ≫ \overline{(s1)} ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}; ∀\underline{x}: Ded ⊳ ∀\underline{x}: \overline{(s1)} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⊢ \overline{(s1)} ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ≫ \overline{(s1)} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⇒ \overline{(s1)} ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}; Gen ⊳ \overline{(s1)} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⇒ \overline{(s1)} ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ≫ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ {ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⇒ \overline{(s1)} ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}; Repetition ⊳ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⇒ \overline{(s1)} ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ≫ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ {ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⇒ \overline{(s1)} ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}; CartProdIsRelation ≫ ∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n; SubsetIsRelation ⊳ ∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⊳ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⇒ \overline{(s1)} ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ≫ ∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ {ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n⌉, p_{0}, c)]

[CrsIsRelation →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n]

[CrsIsRelation →^{tex} “CrsIsRelation”]

[CrsIsRelation →^{pyk} “lemma crsIsRelation”]

CrsIsFunction

[CrsIsFunction →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\} ⊢ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\} ⊢ FromOrderedPair ⊳ \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\} ≫ ˙¬ (\overline{(f1)} = \overline{(crs1)} ⇒ ˙¬ (\overline{(f2)} = \underline{x})n)n; SecondConjunct ⊳ ˙¬ (\overline{(f1)} = \overline{(crs1)} ⇒ ˙¬ (\overline{(f2)} = \underline{x})n)n ≫ \overline{(f2)} = \underline{x}; FromOrderedPair ⊳ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\} ≫ ˙¬ (\overline{(f3)} = \overline{(crs1)} ⇒ ˙¬ (\overline{(f4)} = \underline{x})n)n; SecondConjunct ⊳ ˙¬ (\overline{(f3)} = \overline{(crs1)} ⇒ ˙¬ (\overline{(f4)} = \underline{x})n)n ≫ \overline{(f4)} = \underline{x}; eqSymmetry ⊳ \overline{(f4)} = \underline{x} ≫ \underline{x} = \overline{(f4)}; eqTransitivity ⊳ \overline{(f2)} = \underline{x} ⊳ \underline{x} = \overline{(f4)} ≫ \overline{(f2)} = \overline{(f4)}; ∀\underline{x}: Ded ⊳ ∀\underline{x}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\} ⊢ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\} ⊢ \overline{(f2)} = \overline{(f4)} ≫ \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\} ⇒ \overline{(f2)} = \overline{(f4)}; \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⊢ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⊢ \overline{(f1)} = \overline{(f3)} ⊢ Sep2Formula ⊳ \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ≫ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n)n)n; SecondConjunct ⊳ ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n)n)n ≫ ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n; Sep2Formula ⊳ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ {ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ≫ ˙¬ (\{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (\{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n)n)n; SecondConjunct ⊳ ˙¬ (\{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (\{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n)n)n ≫ ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (\{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n; ExistMP2 ⊳ \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\} ⇒ \overline{(f2)} = \overline{(f4)} ⊳ ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (\{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n ⊳ ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (\{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n ≫ \overline{(f2)} = \overline{(f4)}; ∀\underline{x}: Ded ⊳ ∀\underline{x}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⊢ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⊢ \overline{(f1)} = \overline{(f3)} ⊢ \overline{(f2)} = \overline{(f4)} ≫ ∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)}; CrsIsRelation ≫ ∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n; JoinConjuncts ⊳ ∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⊳ ∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)} ≫ ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n⌉, p_{0}, c)]

[CrsIsFunction →^{stmt} SystemQ ⊢ ∀\underline{x}: ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n]

[CrsIsFunction →^{tex} “CrsIsFunction ”]

[CrsIsFunction →^{pyk} “lemma crsIsFunction”]

CrsIsTotal

[CrsIsTotal →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{x}: λc.TypeRational0(⌈\underline{x}⌉) ⊢⊢ \underline{m} ∈ N ⊢ RationalType ⊳ ⊳ λc.TypeRational0(⌈\underline{x}⌉) ≫ \underline{x} ∈ Q; ToCartProd ⊳ \underline{m} ∈ N ⊳ \underline{x} ∈ Q ≫ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{x}\}\} ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}; eqReflexivity ≫ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{x}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{x}\}\}; IntroExist @ \underline{m} ⊳ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{x}\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{x}\}\} ≫ ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (\{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{x}\}\} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n; Formula2Sep ⊳ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{x}\}\} ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ⊳ ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (\{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{x}\}\} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n ≫ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{x}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\}⌉, p_{0}, c)]

[CrsIsTotal →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{x}: λc.TypeRational0(⌈\underline{x}⌉) ⊢⊢ \underline{m} ∈ N ⊢ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{x}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\}]

[CrsIsTotal →^{tex} “CrsIsTotal”]

[CrsIsTotal →^{pyk} “lemma crsIsTotal”]

CrsIsSeries

[CrsIsSeries →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: \overline{(s1)} ∈ N ⊢ CrsIsTotal ⊳ \overline{(s1)} ∈ N ≫ \{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \underline{x}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\}; IntroExist @ \underline{x} ⊳ \overline{(s1)} ∈ N ≫ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\})n)n; ∀\underline{x}: Ded ⊳ ∀\underline{x}: \overline{(s1)} ∈ N ⊢ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\})n)n ≫ \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\})n)n; Gen ⊳ \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\})n)n ≫ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\})n)n; CrsIsFunction ≫ ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ {ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n; JoinConjuncts ⊳ ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⊳ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ {ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\})n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\})n)n)n)n⌉, p_{0}, c)]

[CrsIsSeries →^{stmt} SystemQ ⊢ ∀\underline{x}: ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\})n)n)n)n]

[CrsIsSeries →^{tex} “CrsIsSeries”]

[CrsIsSeries →^{pyk} “lemma crsIsSeries”]

CrsLookup

[CrsLookup →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{x}: \underline{m} ∈ N ⊢ CrsIsSeries ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\})n)n)n)n; MemberOfSeries ⊳ \underline{m} ∈ N ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\})n)n)n)n ≫ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\}[\underline{m}]\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\}; CrsIsTotal ⊳ \underline{m} ∈ N ≫ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{x}\}\} ∈ {ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\}; eqReflexivity ≫ \underline{m} = \underline{m}; UniqueMember ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\})n)n)n)n ⊳ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\}[\underline{m}]\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⊳ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{x}\}\} ∈ {ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\} ⊳ \underline{m} = \underline{m} ≫ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\}[\underline{m}] = \underline{x}⌉, p_{0}, c)]

[CrsLookup →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{x}: \underline{m} ∈ N ⊢ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} | ˙¬ (∀_{obj}\overline{(crs1)}: ˙¬ (c_{Ph} = \{\{\overline{(crs1)}, \overline{(crs1)}\}, \{\overline{(crs1)}, \underline{x}\}\})n)n\}[\underline{m}] = \underline{x}]

[CrsLookup →^{tex} “CrsLookup”]

[CrsLookup →^{pyk} “lemma crsLookup”]

0f

[0f →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: \underline{m} ∈ N ⊢ CrsLookup ⊳ \underline{m} ∈ N ≫ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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}] = 0; Repetition ⊳ {ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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}] = 0 ≫ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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}] = 0⌉, p_{0}, c)]

[0f →^{stmt} SystemQ ⊢ ∀\underline{m}: \underline{m} ∈ N ⊢ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\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}] = 0]

[0f →^{tex} “0f”]

[0f →^{pyk} “lemma 0f”]

1f

[1f →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: \underline{m} ∈ N ⊢ CrsLookup ⊳ \underline{m} ∈ N ≫ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] = 1; Repetition ⊳ {ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] = 1 ≫ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] = 1⌉, p_{0}, c)]

[1f →^{stmt} SystemQ ⊢ ∀\underline{m}: \underline{m} ∈ N ⊢ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] = 1]

[1f →^{tex} “1f”]

[1f →^{pyk} “lemma 1f”]

ToSingleton

[ToSingleton →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sy)}: \underline{(sx)} = \underline{(sy)} ⊢ WeakenOr1 ⊳ \underline{(sx)} = \underline{(sy)} ≫ ˙¬ (\underline{(sx)} = \underline{(sy)})n ⇒ \underline{(sx)} = \underline{(sy)}; Formula2Pair ⊳ ˙¬ (\underline{(sx)} = \underline{(sy)})n ⇒ \underline{(sx)} = \underline{(sy)} ≫ \underline{(sx)} ∈ \{\underline{(sy)}, \underline{(sy)}\}; Repetition ⊳ \underline{(sx)} ∈ \{\underline{(sy)}, \underline{(sy)}\} ≫ \underline{(sx)} ∈ \{\underline{(sy)}, \underline{(sy)}\}⌉, p_{0}, c)]

[ToSingleton →^{stmt} SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sy)}: \underline{(sx)} = \underline{(sy)} ⊢ \underline{(sx)} ∈ \{\underline{(sy)}, \underline{(sy)}\}]

[ToSingleton →^{tex} “ToSingleton”]

[ToSingleton →^{pyk} “lemma toSingleton”]

FromSameSingleton

[FromSameSingleton →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sy)}: \{\underline{(sx)}, \underline{(sx)}\} = \{\underline{(sy)}, \underline{(sy)}\} ⊢ eqReflexivity ≫ \underline{(sx)} = \underline{(sx)}; ToSingleton ⊳ \underline{(sx)} = \underline{(sx)} ≫ \underline{(sx)} ∈ \{\underline{(sx)}, \underline{(sx)}\}; SENC1 ⊳ \{\underline{(sx)}, \underline{(sx)}\} = \{\underline{(sy)}, \underline{(sy)}\} ⊳ \underline{(sx)} ∈ \{\underline{(sx)}, \underline{(sx)}\} ≫ \underline{(sx)} ∈ \{\underline{(sy)}, \underline{(sy)}\}; FromSingleton ⊳ \underline{(sx)} ∈ \{\underline{(sy)}, \underline{(sy)}\} ≫ \underline{(sx)} = \underline{(sy)}⌉, p_{0}, c)]

[FromSameSingleton →^{stmt} SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sy)}: \{\underline{(sx)}, \underline{(sx)}\} = \{\underline{(sy)}, \underline{(sy)}\} ⊢ \underline{(sx)} = \underline{(sy)}]

[FromSameSingleton →^{tex} “FromSameSingleton”]

[FromSameSingleton →^{pyk} “lemma fromSameSingleton”]

SingletonmembersEqual

[SingletonmembersEqual →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sy)}: ∀\underline{(sz)}: \{\underline{(sx)}, \underline{(sy)}\} = \{\underline{(sz)}, \underline{(sz)}\} ⊢ InPair(1) ≫ \underline{(sx)} ∈ \{\underline{(sx)}, \underline{(sy)}\}; SENC1 ⊳ \{\underline{(sx)}, \underline{(sy)}\} = \{\underline{(sz)}, \underline{(sz)}\} ⊳ \underline{(sx)} ∈ \{\underline{(sx)}, \underline{(sy)}\} ≫ \underline{(sx)} ∈ \{\underline{(sz)}, \underline{(sz)}\}; FromSingleton ⊳ \underline{(sx)} ∈ \{\underline{(sz)}, \underline{(sz)}\} ≫ \underline{(sx)} = \underline{(sz)}; InPair(2) ≫ \underline{(sy)} ∈ \{\underline{(sx)}, \underline{(sy)}\}; SENC1 ⊳ \{\underline{(sx)}, \underline{(sy)}\} = \{\underline{(sz)}, \underline{(sz)}\} ⊳ \underline{(sy)} ∈ \{\underline{(sx)}, \underline{(sy)}\} ≫ \underline{(sy)} ∈ \{\underline{(sz)}, \underline{(sz)}\}; FromSingleton ⊳ \underline{(sy)} ∈ \{\underline{(sz)}, \underline{(sz)}\} ≫ \underline{(sy)} = \underline{(sz)}; eqSymmetry ⊳ \underline{(sy)} = \underline{(sz)} ≫ \underline{(sz)} = \underline{(sy)}; eqTransitivity ⊳ \underline{(sx)} = \underline{(sz)} ⊳ \underline{(sz)} = \underline{(sy)} ≫ \underline{(sx)} = \underline{(sy)}⌉, p_{0}, c)]

[SingletonmembersEqual →^{stmt} SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sy)}: ∀\underline{(sz)}: \{\underline{(sx)}, \underline{(sy)}\} = \{\underline{(sz)}, \underline{(sz)}\} ⊢ \underline{(sx)} = \underline{(sy)}]

[SingletonmembersEqual →^{tex} “SingletonmembersEqual”]

[SingletonmembersEqual →^{pyk} “lemma singletonmembersEqual”]

UnequalsNotInSingleton

[UnequalsNotInSingleton →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sy)}: ∀\underline{(sz)}: \{\underline{(sx)}, \underline{(sy)}\} = \{\underline{(sz)}, \underline{(sz)}\} ⊢ SingletonmembersEqual ⊳ \{\underline{(sx)}, \underline{(sy)}\} = \{\underline{(sz)}, \underline{(sz)}\} ≫ \underline{(sx)} = \underline{(sy)}; ∀\underline{(sx)}: ∀\underline{(sy)}: ∀\underline{(sz)}: Ded ⊳ ∀\underline{(sx)}: ∀\underline{(sy)}: ∀\underline{(sz)}: \{\underline{(sx)}, \underline{(sy)}\} = \{\underline{(sz)}, \underline{(sz)}\} ⊢ \underline{(sx)} = \underline{(sy)} ≫ \{\underline{(sx)}, \underline{(sy)}\} = \{\underline{(sz)}, \underline{(sz)}\} ⇒ \underline{(sx)} = \underline{(sy)}; ˙¬ (\underline{(sx)} = \underline{(sy)})n ⊢ MT ⊳ \{\underline{(sx)}, \underline{(sy)}\} = \{\underline{(sz)}, \underline{(sz)}\} ⇒ \underline{(sx)} = \underline{(sy)} ⊳ ˙¬ (\underline{(sx)} = \underline{(sy)})n ≫ ˙¬ (\{\underline{(sx)}, \underline{(sy)}\} = \{\underline{(sz)}, \underline{(sz)}\})n⌉, p_{0}, c)]

[UnequalsNotInSingleton →^{stmt} SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sy)}: ∀\underline{(sz)}: ˙¬ (\underline{(sx)} = \underline{(sy)})n ⊢ ˙¬ (\{\underline{(sx)}, \underline{(sy)}\} = \{\underline{(sz)}, \underline{(sz)}\})n]

[UnequalsNotInSingleton →^{tex} “UnequalsNotInSingleton”]

[UnequalsNotInSingleton →^{pyk} “lemma unequalsNotInSingleton”]

NonsingletonmembersUnequal

[NonsingletonmembersUnequal →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sy)}: \underline{(sx)} = \underline{(sy)} ⊢ eqReflexivity ≫ \underline{(sx)} = \underline{(sx)}; SamePair ⊳ \underline{(sx)} = \underline{(sx)} ⊳ \underline{(sx)} = \underline{(sy)} ≫ \{\underline{(sx)}, \underline{(sx)}\} = \{\underline{(sx)}, \underline{(sy)}\}; Repetition ⊳ \{\underline{(sx)}, \underline{(sx)}\} = \{\underline{(sx)}, \underline{(sy)}\} ≫ \{\underline{(sx)}, \underline{(sx)}\} = \{\underline{(sx)}, \underline{(sy)}\}; eqSymmetry ⊳ \{\underline{(sx)}, \underline{(sx)}\} = \{\underline{(sx)}, \underline{(sy)}\} ≫ \{\underline{(sx)}, \underline{(sy)}\} = \{\underline{(sx)}, \underline{(sx)}\}; ∀\underline{(sx)}: ∀\underline{(sy)}: Ded ⊳ ∀\underline{(sx)}: ∀\underline{(sy)}: \underline{(sx)} = \underline{(sy)} ⊢ \{\underline{(sx)}, \underline{(sy)}\} = \{\underline{(sx)}, \underline{(sx)}\} ≫ \underline{(sx)} = \underline{(sy)} ⇒ \{\underline{(sx)}, \underline{(sy)}\} = \{\underline{(sx)}, \underline{(sx)}\}; ˙¬ (\{\underline{(sx)}, \underline{(sy)}\} = \{\underline{(sx)}, \underline{(sx)}\})n ⊢ MT⊳\underline{(sx)} = \underline{(sy)} ⇒ \{\underline{(sx)}, \underline{(sy)}\} = \{\underline{(sx)}, \underline{(sx)}\} ⊳ ˙¬ (\{\underline{(sx)}, \underline{(sy)}\} = \{\underline{(sx)}, \underline{(sx)}\})n ≫ ˙¬ (\underline{(sx)} = \underline{(sy)})n⌉, p_{0}, c)]

[NonsingletonmembersUnequal →^{stmt} SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sy)}: ˙¬ (\{\underline{(sx)}, \underline{(sy)}\} = \{\underline{(sx)}, \underline{(sx)}\})n ⊢ ˙¬ (\underline{(sx)} = \underline{(sy)})n]

[NonsingletonmembersUnequal →^{tex} “NonsingletonmembersUnequal”]

[NonsingletonmembersUnequal →^{pyk} “lemma nonsingletonmembersUnequal”]

FromOrderedPair

[FromOrderedPair →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sx1)}: ∀\underline{(sy)}: ∀\underline{(sy1)}: \underline{(sx1)} = \underline{(sy1)} ⊢ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sy1)}\}\} ⊢ Repetition ⊳ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sy1)}\}\} ≫ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sy1)}\}\}; eqReflexivity ≫ \underline{(sx1)} = \underline{(sx1)}; SamePair ⊳ \underline{(sx1)} = \underline{(sx1)} ⊳ \underline{(sx1)} = \underline{(sy1)} ≫ \{\underline{(sx1)}, \underline{(sx1)}\} = \{\underline{(sx1)}, \underline{(sy1)}\}; Repetition ⊳ \{\underline{(sx1)}, \underline{(sx1)}\} = \{\underline{(sx1)}, \underline{(sy1)}\} ≫ \{\underline{(sx1)}, \underline{(sx1)}\} = \{\underline{(sx1)}, \underline{(sy1)}\}; eqReflexivity ≫ \{\underline{(sx1)}, \underline{(sx1)}\} = \{\underline{(sx1)}, \underline{(sx1)}\}; SamePair ⊳ \{\underline{(sx1)}, \underline{(sx1)}\} = \{\underline{(sx1)}, \underline{(sx1)}\} ⊳ \{\underline{(sx1)}, \underline{(sx1)}\} = \{\underline{(sx1)}, \underline{(sy1)}\} ≫ \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sx1)}\}\} = \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sy1)}\}\}; Repetition ⊳ \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sx1)}\}\} = \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sy1)}\}\} ≫ \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sx1)}\}\} = \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sy1)}\}\}; eqSymmetry ⊳ \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sx1)}\}\} = \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sy1)}\}\} ≫ \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sy1)}\}\} = \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sx1)}\}\}; eqTransitivity ⊳ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sy1)}\}\} ⊳ \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sy1)}\}\} = \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sx1)}\}\} ≫ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sx1)}\}\}; InPair(1) ≫ \{\underline{(sx)}, \underline{(sx)}\} ∈ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\}; SENC1 ⊳ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sx1)}\}\} ⊳ \{\underline{(sx)}, \underline{(sx)}\} ∈ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ≫ \{\underline{(sx)}, \underline{(sx)}\} ∈ \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sx1)}\}\}; FromSingleton ⊳ \{\underline{(sx)}, \underline{(sx)}\} ∈ \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sx1)}\}\} ≫ \{\underline{(sx)}, \underline{(sx)}\} = \{\underline{(sx1)}, \underline{(sx1)}\}; FromSameSingleton ⊳ \{\underline{(sx)}, \underline{(sx)}\} = \{\underline{(sx1)}, \underline{(sx1)}\} ≫ \underline{(sx)} = \underline{(sx1)}; eqSymmetry ⊳ \{\underline{(sx)}, \underline{(sx)}\} = \{\underline{(sx1)}, \underline{(sx1)}\} ≫ \{\underline{(sx1)}, \underline{(sx1)}\} = \{\underline{(sx)}, \underline{(sx)}\}; SameSingleton ⊳ \{\underline{(sx1)}, \underline{(sx1)}\} = \{\underline{(sx)}, \underline{(sx)}\} ≫ \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sx1)}\}\} = \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sx)}\}\}; eqTransitivity ⊳ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sx1)}\}\} ⊳ \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sx1)}\}\} = \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sx)}\}\} ≫ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sx)}\}\}; InPair(2) ≫ \{\underline{(sx)}, \underline{(sy)}\} ∈ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\}; SENC1 ⊳ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sx)}\}\} ⊳ \{\underline{(sx)}, \underline{(sy)}\} ∈ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ≫ \{\underline{(sx)}, \underline{(sy)}\} ∈ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sx)}\}\}; FromSingleton ⊳ \{\underline{(sx)}, \underline{(sy)}\} ∈ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sx)}\}\} ≫ \{\underline{(sx)}, \underline{(sy)}\} = \{\underline{(sx)}, \underline{(sx)}\}; SingletonmembersEqual ⊳ \{\underline{(sx)}, \underline{(sy)}\} = \{\underline{(sx)}, \underline{(sx)}\} ≫ \underline{(sx)} = \underline{(sy)}; eqSymmetry ⊳ \underline{(sx)} = \underline{(sy)} ≫ \underline{(sy)} = \underline{(sx)}; eqTransitivity4 ⊳ \underline{(sy)} = \underline{(sx)} ⊳ \underline{(sx)} = \underline{(sx1)} ⊳ \underline{(sx1)} = \underline{(sy1)} ≫ \underline{(sy)} = \underline{(sy1)}; JoinConjuncts ⊳ \underline{(sx)} = \underline{(sx1)} ⊳ \underline{(sy)} = \underline{(sy1)} ≫ ˙¬ (\underline{(sx)} = \underline{(sx1)} ⇒ ˙¬ (\underline{(sy)} = \underline{(sy1)})n)n; ∀\underline{(sx)}: ∀\underline{(sx1)}: ∀\underline{(sy)}: ∀\underline{(sy1)}: ˙¬ (\underline{(sx1)} = \underline{(sy1)})n ⊢ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sy1)}\}\} ⊢ Repetition ⊳ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sy1)}\}\} ≫ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sy1)}\}\}; InPair(1) ≫ \{\underline{(sx)}, \underline{(sx)}\} ∈ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\}; SENC1 ⊳ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sy1)}\}\} ⊳ \{\underline{(sx)}, \underline{(sx)}\} ∈ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ≫ \{\underline{(sx)}, \underline{(sx)}\} ∈ \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sy1)}\}\}; Pair2Formula ⊳ \{\underline{(sx)}, \underline{(sx)}\} ∈ \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sy1)}\}\} ≫ ˙¬ (\{\underline{(sx)}, \underline{(sx)}\} = \{\underline{(sx1)}, \underline{(sx1)}\})n ⇒ \{\underline{(sx)}, \underline{(sx)}\} = \{\underline{(sx1)}, \underline{(sy1)}\}; UnequalsNotInSingleton ⊳ ˙¬ (\underline{(sx1)} = \underline{(sy1)})n ≫ ˙¬ (\{\underline{(sx1)}, \underline{(sy1)}\} = \{\underline{(sx)}, \underline{(sx)}\})n; NeqSymmetry ⊳ ˙¬ (\{\underline{(sx1)}, \underline{(sy1)}\} = \{\underline{(sx)}, \underline{(sx)}\})n ≫ ˙¬ (\{\underline{(sx)}, \underline{(sx)}\} = \{\underline{(sx1)}, \underline{(sy1)}\})n; NegateDisjunct2 ⊳ ˙¬ (\{\underline{(sx)}, \underline{(sx)}\} = \{\underline{(sx1)}, \underline{(sx1)}\})n ⇒ \{\underline{(sx)}, \underline{(sx)}\} = \{\underline{(sx1)}, \underline{(sy1)}\}⊳ ˙¬ (\{\underline{(sx)}, \underline{(sx)}\} = \{\underline{(sx1)}, \underline{(sy1)}\})n ≫ \{\underline{(sx)}, \underline{(sx)}\} = \{\underline{(sx1)}, \underline{(sx1)}\}; FromSameSingleton ⊳ \{\underline{(sx)}, \underline{(sx)}\} = \{\underline{(sx1)}, \underline{(sx1)}\} ≫ \underline{(sx)} = \underline{(sx1)}; InPair(2) ≫ \{\underline{(sx1)}, \underline{(sy1)}\} ∈ \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sy1)}\}\}; SENC2 ⊳ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sy1)}\}\} ⊳ \{\underline{(sx1)}, \underline{(sy1)}\} ∈ \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sy1)}\}\} ≫ \{\underline{(sx1)}, \underline{(sy1)}\} ∈ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\}; Pair2Formula ⊳ \{\underline{(sx1)}, \underline{(sy1)}\} ∈ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ≫ ˙¬ (\{\underline{(sx1)}, \underline{(sy1)}\} = \{\underline{(sx)}, \underline{(sx)}\})n ⇒ \{\underline{(sx1)}, \underline{(sy1)}\} = \{\underline{(sx)}, \underline{(sy)}\}; NegateDisjunct1 ⊳ ˙¬ (\{\underline{(sx1)}, \underline{(sy1)}\} = \{\underline{(sx)}, \underline{(sx)}\})n ⇒ \{\underline{(sx1)}, \underline{(sy1)}\} = \{\underline{(sx)}, \underline{(sy)}\} ⊳ ˙¬ (\{\underline{(sx1)}, \underline{(sy1)}\} = \{\underline{(sx)}, \underline{(sx)}\})n ≫ \{\underline{(sx1)}, \underline{(sy1)}\} = \{\underline{(sx)}, \underline{(sy)}\}; InPair(2) ≫ \underline{(sy)} ∈ \{\underline{(sx)}, \underline{(sy)}\}; SENC2 ⊳ \{\underline{(sx1)}, \underline{(sy1)}\} = \{\underline{(sx)}, \underline{(sy)}\} ⊳ \underline{(sy)} ∈ \{\underline{(sx)}, \underline{(sy)}\} ≫ \underline{(sy)} ∈ \{\underline{(sx1)}, \underline{(sy1)}\}; Pair2Formula ⊳ \underline{(sy)} ∈ \{\underline{(sx1)}, \underline{(sy1)}\} ≫ ˙¬ (\underline{(sy)} = \underline{(sx1)})n ⇒ \underline{(sy)} = \underline{(sy1)}; UnequalsNotInSingleton ⊳ ˙¬ (\underline{(sx1)} = \underline{(sy1)})n ≫ ˙¬ (\{\underline{(sx1)}, \underline{(sy1)}\} = \{\underline{(sx)}, \underline{(sx)}\})n; SubNeqLeft ⊳ \{\underline{(sx1)}, \underline{(sy1)}\} = \{\underline{(sx)}, \underline{(sy)}\} ⊳ ˙¬ (\{\underline{(sx1)}, \underline{(sy1)}\} = \{\underline{(sx)}, \underline{(sx)}\})n ≫ ˙¬ (\{\underline{(sx)}, \underline{(sy)}\} = \{\underline{(sx)}, \underline{(sx)}\})n; NonsingletonmembersUnequal ⊳ ˙¬ (\{\underline{(sx)}, \underline{(sy)}\} = \{\underline{(sx)}, \underline{(sx)}\})n ≫ ˙¬ (\underline{(sx)} = \underline{(sy)})n; SubNeqLeft ⊳ \underline{(sx)} = \underline{(sx1)} ⊳ ˙¬ (\underline{(sx)} = \underline{(sy)})n ≫ ˙¬ (\underline{(sx1)} = \underline{(sy)})n; NeqSymmetry ⊳ ˙¬ (\underline{(sx1)} = \underline{(sy)})n ≫ ˙¬ (\underline{(sy)} = \underline{(sx1)})n; NegateDisjunct1 ⊳ ˙¬ (\underline{(sy)} = \underline{(sx1)})n ⇒ \underline{(sy)} = \underline{(sy1)} ⊳ ˙¬ (\underline{(sy)} = \underline{(sx1)})n ≫ \underline{(sy)} = \underline{(sy1)}; JoinConjuncts ⊳ \underline{(sx)} = \underline{(sx1)} ⊳ \underline{(sy)} = \underline{(sy1)} ≫ ˙¬ (\underline{(sx)} = \underline{(sx1)} ⇒ ˙¬ (\underline{(sy)} = \underline{(sy1)})n)n; ∀\underline{(sx)}: ∀\underline{(sx1)}: ∀\underline{(sy)}: ∀\underline{(sy1)}: Ded ⊳ ∀\underline{(sx)}: ∀\underline{(sx1)}: ∀\underline{(sy)}: ∀\underline{(sy1)}: \underline{(sx1)} = \underline{(sy1)} ⊢ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sy1)}\}\} ⊢ ˙¬ (\underline{(sx)} = \underline{(sx1)} ⇒ ˙¬ (\underline{(sy)} = \underline{(sy1)})n)n ≫ \underline{(sx1)} = \underline{(sy1)} ⇒ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sy1)}\}\} ⇒ ˙¬ (\underline{(sx)} = \underline{(sx1)} ⇒ ˙¬ (\underline{(sy)} = \underline{(sy1)})n)n; Ded ⊳ ∀\underline{(sx)}: ∀\underline{(sx1)}: ∀\underline{(sy)}: ∀\underline{(sy1)}: ˙¬ (\underline{(sx1)} = \underline{(sy1)})n ⊢ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sy1)}\}\} ⊢ ˙¬ (\underline{(sx)} = \underline{(sx1)} ⇒ ˙¬ (\underline{(sy)} = \underline{(sy1)})n)n ≫ ˙¬ (\underline{(sx1)} = \underline{(sy1)})n ⇒ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sy1)}\}\} ⇒ ˙¬ (\underline{(sx)} = \underline{(sx1)} ⇒ ˙¬ (\underline{(sy)} = \underline{(sy1)})n)n; \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sy1)}\}\} ⊢ FromNegations ⊳ \underline{(sx1)} = \underline{(sy1)} ⇒ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sy1)}\}\} ⇒ ˙¬ (\underline{(sx)} = \underline{(sx1)} ⇒ ˙¬ (\underline{(sy)} = \underline{(sy1)})n)n ⊳ ˙¬ (\underline{(sx1)} = \underline{(sy1)})n ⇒ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sy1)}\}\} ⇒ ˙¬ (\underline{(sx)} = \underline{(sx1)} ⇒ ˙¬ (\underline{(sy)} = \underline{(sy1)})n)n ≫ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sy1)}\}\} ⇒ ˙¬ (\underline{(sx)} = \underline{(sx1)} ⇒ ˙¬ (\underline{(sy)} = \underline{(sy1)})n)n; MP ⊳ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sy1)}\}\} ⇒ ˙¬ (\underline{(sx)} = \underline{(sx1)} ⇒ ˙¬ (\underline{(sy)} = \underline{(sy1)})n)n ⊳ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sy1)}\}\} ≫ ˙¬ (\underline{(sx)} = \underline{(sx1)} ⇒ ˙¬ (\underline{(sy)} = \underline{(sy1)})n)n⌉, p_{0}, c)]

[FromOrderedPair →^{stmt} SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sx1)}: ∀\underline{(sy)}: ∀\underline{(sy1)}: \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sy1)}\}\} ⊢ ˙¬ (\underline{(sx)} = \underline{(sx1)} ⇒ ˙¬ (\underline{(sy)} = \underline{(sy1)})n)n]

[FromOrderedPair →^{tex} “FromOrderedPair”]

[FromOrderedPair →^{pyk} “lemma fromOrderedPair”]

FromOrderedPair(1)

[FromOrderedPair(1) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sx1)}: ∀\underline{(sy)}: ∀\underline{(sy1)}: \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sy1)}\}\} ⊢ FromOrderedPair ⊳ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sy1)}\}\} ≫ ˙¬ (\underline{(sx)} = \underline{(sx1)} ⇒ ˙¬ (\underline{(sy)} = \underline{(sy1)})n)n; FirstConjunct ⊳ ˙¬ (\underline{(sx)} = \underline{(sx1)} ⇒ ˙¬ (\underline{(sy)} = \underline{(sy1)})n)n ≫ \underline{(sx)} = \underline{(sx1)}⌉, p_{0}, c)]

[FromOrderedPair(1) →^{stmt} SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sx1)}: ∀\underline{(sy)}: ∀\underline{(sy1)}: \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sy1)}\}\} ⊢ \underline{(sx)} = \underline{(sx1)}]

[FromOrderedPair(1) →^{tex} “FromOrderedPair(1)”]

[FromOrderedPair(1) →^{pyk} “lemma fromOrderedPair(1)”]

FromOrderedPair(2)

[FromOrderedPair(2) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sx1)}: ∀\underline{(sy)}: ∀\underline{(sy1)}: \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sy1)}\}\} ⊢ FromOrderedPair ⊳ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sy1)}\}\} ≫ ˙¬ (\underline{(sx)} = \underline{(sx1)} ⇒ ˙¬ (\underline{(sy)} = \underline{(sy1)})n)n; SecondConjunct ⊳ ˙¬ (\underline{(sx)} = \underline{(sx1)} ⇒ ˙¬ (\underline{(sy)} = \underline{(sy1)})n)n ≫ \underline{(sy)} = \underline{(sy1)}⌉, p_{0}, c)]

[FromOrderedPair(2) →^{stmt} SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sx1)}: ∀\underline{(sy)}: ∀\underline{(sy1)}: \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sy1)}\}\} ⊢ \underline{(sy)} = \underline{(sy1)}]

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

[FromOrderedPair(2) →^{pyk} “lemma fromOrderedPair(2)”]

FromCartProd

[FromCartProd →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sx1)}: ∀\underline{(sy)}: ∀\underline{(sy1)}: \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ∈ \{ph ∈ P(P(Union(\{\underline{(sx1)}, \underline{(sy1)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx1)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy1)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ⊢ Repetition ⊳ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ∈ \{ph ∈ P(P(Union(\{\underline{(sx1)}, \underline{(sy1)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx1)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy1)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ≫ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ∈ \{ph ∈ P(P(Union(\{\underline{(sx1)}, \underline{(sy1)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx1)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy1)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}; Sep2Formula ⊳ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ∈ \{ph ∈ P(P(Union(\{\underline{(sx1)}, \underline{(sy1)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx1)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy1)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ≫ ˙¬ (\{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ∈ P(P(Union(\{\underline{(sx1)}, \underline{(sy1)}\}))) ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx1)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy1)})n)n ⇒ ˙¬ (\{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n)n)n; SecondConjunct ⊳ ˙¬ (\{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ∈ P(P(Union(\{\underline{(sx1)}, \underline{(sy1)}\}))) ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx1)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy1)})n)n ⇒ ˙¬ (\{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n)n)n ≫ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx1)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy1)})n)n ⇒ ˙¬ (\{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n; Repetition ⊳ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx1)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy1)})n)n ⇒ ˙¬ (\{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ≫ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx1)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy1)})n)n ⇒ ˙¬ (\{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n; ∀\underline{(sx)}: ∀\underline{(sx1)}: ∀\underline{(sy)}: ∀\underline{(sy1)}: ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx1)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy1)})n)n ⇒ ˙¬ (\{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n ⊢ FirstConjunct ⊳ ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx1)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy1)})n)n ⇒ ˙¬ (\{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n ≫ ˙¬ (\overline{(op1)} ∈ \underline{(sx1)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy1)})n)n; FirstConjunct ⊳ ˙¬ (\overline{(op1)} ∈ \underline{(sx1)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy1)})n)n ≫ \overline{(op1)} ∈ \underline{(sx1)}; SecondConjunct ⊳ ˙¬ (\overline{(op1)} ∈ \underline{(sx1)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy1)})n)n ≫ \overline{(op2)} ∈ \underline{(sy1)}; SecondConjunct ⊳ ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx1)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy1)})n)n ⇒ ˙¬ (\{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n ≫ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\}; FromOrderedPair(1) ⊳ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\} ≫ \underline{(sx)} = \overline{(op1)}; SameMember(2) ⊳ \underline{(sx)} = \overline{(op1)} ⊳ \overline{(op1)} ∈ \underline{(sx1)} ≫ \underline{(sx)} ∈ \underline{(sx1)}; FromOrderedPair(2) ⊳ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\} ≫ \underline{(sy)} = \overline{(op2)}; SameMember(2) ⊳ \underline{(sy)} = \overline{(op2)} ⊳ \overline{(op2)} ∈ \underline{(sy1)} ≫ \underline{(sy)} ∈ \underline{(sy1)}; JoinConjuncts ⊳ \underline{(sx)} ∈ \underline{(sx1)} ⊳ \underline{(sy)} ∈ \underline{(sy1)} ≫ ˙¬ (\underline{(sx)} ∈ \underline{(sx1)} ⇒ ˙¬ (\underline{(sy)} ∈ \underline{(sy1)})n)n; Ded ⊳ ∀\underline{(sx)}: ∀\underline{(sx1)}: ∀\underline{(sy)}: ∀\underline{(sy1)}: ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx1)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy1)})n)n ⇒ ˙¬ (\{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n ⊢ ˙¬ (\underline{(sx)} ∈ \underline{(sx1)} ⇒ ˙¬ (\underline{(sy)} ∈ \underline{(sy1)})n)n ≫ ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx1)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy1)})n)n ⇒ ˙¬ (\{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n ⇒ ˙¬ (\underline{(sx)} ∈ \underline{(sx1)} ⇒ ˙¬ (\underline{(sy)} ∈ \underline{(sy1)})n)n; TwiceExistMP ⊳ ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx1)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy1)})n)n ⇒ ˙¬ (\{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n ⇒ ˙¬ (\underline{(sx)} ∈ \underline{(sx1)} ⇒ ˙¬ (\underline{(sy)} ∈ \underline{(sy1)})n)n ⊳ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx1)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy1)})n)n ⇒ ˙¬ (\{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ≫ ˙¬ (\underline{(sx)} ∈ \underline{(sx1)} ⇒ ˙¬ (\underline{(sy)} ∈ \underline{(sy1)})n)n⌉, p_{0}, c)]

[FromCartProd →^{stmt} SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sx1)}: ∀\underline{(sy)}: ∀\underline{(sy1)}: \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ∈ \{ph ∈ P(P(Union(\{\underline{(sx1)}, \underline{(sy1)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx1)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy1)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ⊢ ˙¬ (\underline{(sx)} ∈ \underline{(sx1)} ⇒ ˙¬ (\underline{(sy)} ∈ \underline{(sy1)})n)n]

[FromCartProd →^{tex} “FromCartProd”]

[FromCartProd →^{pyk} “lemma fromCartProd”]

FromCartProd(1)

[FromCartProd(1) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sx1)}: ∀\underline{(sy)}: ∀\underline{(sy1)}: \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ∈ \{ph ∈ P(P(Union(\{\underline{(sx1)}, \underline{(sy1)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx1)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy1)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ⊢ FromCartProd ⊳ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ∈ \{ph ∈ P(P(Union(\{\underline{(sx1)}, \underline{(sy1)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx1)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy1)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ≫ ˙¬ (\underline{(sx)} ∈ \underline{(sx1)} ⇒ ˙¬ (\underline{(sy)} ∈ \underline{(sy1)})n)n; FirstConjunct ⊳ ˙¬ (\underline{(sx)} ∈ \underline{(sx1)} ⇒ ˙¬ (\underline{(sy)} ∈ \underline{(sy1)})n)n ≫ \underline{(sx)} ∈ \underline{(sx1)}⌉, p_{0}, c)]

[FromCartProd(1) →^{stmt} SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sx1)}: ∀\underline{(sy)}: ∀\underline{(sy1)}: \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ∈ \{ph ∈ P(P(Union(\{\underline{(sx1)}, \underline{(sy1)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx1)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy1)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ⊢ \underline{(sx)} ∈ \underline{(sx1)}] [FromCartProd(1) →^{tex} “FromCartProd(1)”]

[FromCartProd(1) →^{pyk} “lemma fromCartProd(1)”]

FromCartProd(2)

[FromCartProd(2) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sx1)}: ∀\underline{(sy)}: ∀\underline{(sy1)}: \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ∈ \{ph ∈ P(P(Union(\{\underline{(sx1)}, \underline{(sy1)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx1)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy1)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ⊢ FromCartProd ⊳ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ∈ \{ph ∈ P(P(Union(\{\underline{(sx1)}, \underline{(sy1)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx1)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy1)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ≫ ˙¬ (\underline{(sx)} ∈ \underline{(sx1)} ⇒ ˙¬ (\underline{(sy)} ∈ \underline{(sy1)})n)n; SecondConjunct ⊳ ˙¬ (\underline{(sx)} ∈ \underline{(sx1)} ⇒ ˙¬ (\underline{(sy)} ∈ \underline{(sy1)})n)n ≫ \underline{(sy)} ∈ \underline{(sy1)}⌉, p_{0}, c)]

[FromCartProd(2) →^{stmt} SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sx1)}: ∀\underline{(sy)}: ∀\underline{(sy1)}: \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} ∈ \{ph ∈ P(P(Union(\{\underline{(sx1)}, \underline{(sy1)}\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ \underline{(sx1)} ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sy1)})n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ⊢ \underline{(sy)} ∈ \underline{(sy1)}]

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

[FromCartProd(2) →^{pyk} “lemma fromCartProd(2)”]

sameOrderedPair

[sameOrderedPair →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sx1)}: ∀\underline{(sy)}: ∀\underline{(sy1)}: \underline{(sx)} = \underline{(sx1)} ⊢ \underline{(sy)} = \underline{(sy1)} ⊢ SameSingleton ⊳ \underline{(sx)} = \underline{(sx1)} ≫ \{\underline{(sx)}, \underline{(sx)}\} = \{\underline{(sx1)}, \underline{(sx1)}\}; SamePair ⊳ \underline{(sx)} = \underline{(sx1)} ⊳ \underline{(sy)} = \underline{(sy1)} ≫ \{\underline{(sx)}, \underline{(sy)}\} = \{\underline{(sx1)}, \underline{(sy1)}\}; SamePair ⊳ \{\underline{(sx)}, \underline{(sx)}\} = \{\underline{(sx1)}, \underline{(sx1)}\} ⊳ \{\underline{(sx)}, \underline{(sy)}\} = \{\underline{(sx1)}, \underline{(sy1)}\} ≫ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sy1)}\}\}; Repetition ⊳ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sy1)}\}\} ≫ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sy1)}\}\}⌉, p_{0}, c)]

[sameOrderedPair →^{stmt} SystemQ ⊢ ∀\underline{(sx)}: ∀\underline{(sx1)}: ∀\underline{(sy)}: ∀\underline{(sy1)}: \underline{(sx)} = \underline{(sx1)} ⊢ \underline{(sy)} = \underline{(sy1)} ⊢ \{\{\underline{(sx)}, \underline{(sx)}\}, \{\underline{(sx)}, \underline{(sy)}\}\} = \{\{\underline{(sx1)}, \underline{(sx1)}\}, \{\underline{(sx1)}, \underline{(sy1)}\}\}]

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

InSeriesHelper

[InSeriesHelper →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(sx)}: ∀\underline{(sy)}: \underline{(sy)} ∈ \underline{(fx)} ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sx)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ⊢ ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sx)})n)n ⇒ ˙¬ (\underline{(sy)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n ⊢ FirstConjunct ⊳ ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sx)})n)n ⇒ ˙¬ (\underline{(sy)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n ≫ ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sx)})n)n; FirstConjunct ⊳ ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sx)})n)n ≫ \overline{(op1)} ∈ N; SecondConjunct ⊳ ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sx)})n)n ⇒ ˙¬ (\underline{(sy)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n ≫ \underline{(sy)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\}; SameMember ⊳ \underline{(sy)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\} ⊳ \underline{(sy)} ∈ \underline{(fx)} ≫ \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\} ∈ \underline{(fx)}; MemberOfSeries ⊳ \overline{(op1)} ∈ N ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sx)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ≫ \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \underline{(fx)}[\overline{(op1)}]\}\} ∈ \underline{(fx)}; eqReflexivity ≫ \overline{(op1)} = \overline{(op1)}; UniqueMember ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sx)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ⊳ \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\} ∈ \underline{(fx)} ⊳ \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \underline{(fx)}[\overline{(op1)}]\}\} ∈ \underline{(fx)} ⊳ \overline{(op1)} = \overline{(op1)} ≫ \overline{(op2)} = \underline{(fx)}[\overline{(op1)}]; sameOrderedPair ⊳ \overline{(op1)} = \overline{(op1)} ⊳ \overline{(op2)} = \underline{(fx)}[\overline{(op1)}] ≫ \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \underline{(fx)}[\overline{(op1)}]\}\}; eqTransitivity ⊳ \underline{(sy)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\} ⊳ \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \underline{(fx)}[\overline{(op1)}]\}\} ≫ \underline{(sy)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \underline{(fx)}[\overline{(op1)}]\}\}; IntroExist @ \overline{(op1)} ⊳ \underline{(sy)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \underline{(fx)}[\overline{(op1)}]\}\} ≫ ˙¬ (∀_{obj}\underline{m}: ˙¬ (\underline{(sy)} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fx)}[\underline{m}]\}\})n)n; ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(sx)}: ∀\underline{(sy)}: Ded ⊳ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(sx)}: ∀\underline{(sy)}: \underline{(sy)} ∈ \underline{(fx)} ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sx)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ⊢ ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sx)})n)n ⇒ ˙¬ (\underline{(sy)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n ⊢ ˙¬ (∀_{obj}\underline{m}: ˙¬ (\underline{(sy)} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fx)}[\underline{m}]\}\})n)n ≫ \underline{(sy)} ∈ \underline{(fx)} ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sx)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ⇒ ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sx)})n)n ⇒ ˙¬ (\underline{(sy)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n ⇒ ˙¬ (∀_{obj}\underline{m}: ˙¬ (\underline{(sy)} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fx)}[\underline{m}]\}\})n)n⌉, p_{0}, c)]

[InSeriesHelper →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(sx)}: ∀\underline{(sy)}: \underline{(sy)} ∈ \underline{(fx)} ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sx)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ⇒ ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sx)})n)n ⇒ ˙¬ (\underline{(sy)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n ⇒ ˙¬ (∀_{obj}\underline{m}: ˙¬ (\underline{(sy)} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fx)}[\underline{m}]\}\})n)n]

[InSeriesHelper →^{tex} “InSeriesHelper”]

[InSeriesHelper →^{pyk} “lemma inSeries helper”]

InSeries

[InSeries →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(sx)}: ∀\underline{(sy)}: λc.Typeseries0(⌈\underline{(fx)}⌉, ⌈\underline{(sx)}⌉) ⊢⊢ \underline{(sy)} ∈ \underline{(fx)} ⊢ SeriesType ⊳ ⊳ λc.Typeseries0(⌈\underline{(fx)}⌉, ⌈\underline{(sx)}⌉) ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sx)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n; Repetition ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sx)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sx)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n; FirstConjunct ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sx)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ≫ ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sx)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n; Repetition ⊳ ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sx)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ≫ ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sx)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n; FirstConjunct ⊳ ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sx)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ≫ ∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sx)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n; Repetition ⊳ ∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sx)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ≫ ∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sx)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n; A4 @ \underline{(sy)}⊳∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sx)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ≫ \underline{(sy)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sx)})n)n ⇒ ˙¬ (\underline{(sy)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n; MP ⊳ \underline{(sy)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sx)})n)n ⇒ ˙¬ (\underline{(sy)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⊳ \underline{(sy)} ∈ \underline{(fx)} ≫ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sx)})n)n ⇒ ˙¬ (\underline{(sy)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n; Repetition ⊳ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sx)})n)n ⇒ ˙¬ (\underline{(sy)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ≫ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sx)})n)n ⇒ ˙¬ (\underline{(sy)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n; InSeriesHelper ≫ \underline{(sy)} ∈ \underline{(fx)} ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sx)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ⇒ ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sx)})n)n ⇒ ˙¬ (\underline{(sy)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n ⇒ ˙¬ (∀_{obj}\underline{m}: ˙¬ (\underline{(sy)} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fx)}[\underline{m}]\}\})n)n; MP2 ⊳ \underline{(sy)} ∈ \underline{(fx)} ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sx)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ⇒ ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sx)})n)n ⇒ ˙¬ (\underline{(sy)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n ⇒ ˙¬ (∀_{obj}\underline{m}: ˙¬ (\underline{(sy)} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fx)}[\underline{m}]\}\})n)n ⊳ \underline{(sy)} ∈ \underline{(fx)} ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sx)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ≫ ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sx)})n)n ⇒ ˙¬ (\underline{(sy)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n ⇒ ˙¬ (∀_{obj}\underline{m}: ˙¬ (\underline{(sy)} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fx)}[\underline{m}]\}\})n)n; TwiceExistMP ⊳ ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sx)})n)n ⇒ ˙¬ (\underline{(sy)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n ⇒ ˙¬ (∀_{obj}\underline{m}: ˙¬ (\underline{(sy)} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fx)}[\underline{m}]\}\})n)n ⊳ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sx)})n)n ⇒ ˙¬ (\underline{(sy)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ≫ ˙¬ (∀_{obj}\underline{m}: ˙¬ (\underline{(sy)} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fx)}[\underline{m}]\}\})n)n⌉, p_{0}, c)]

[InSeries →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(sx)}: ∀\underline{(sy)}: λc.Typeseries0(⌈\underline{(fx)}⌉, ⌈\underline{(sx)}⌉) ⊢⊢ \underline{(sy)} ∈ \underline{(fx)} ⊢ ˙¬ (∀_{obj}\underline{m}: ˙¬ (\underline{(sy)} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fx)}[\underline{m}]\}\})n)n]

[InSeries →^{tex} “InSeries”]

[InSeries →^{pyk} “lemma inSeries”]

To = f(Subset)(Helper)

[To = f(Subset)(Helper) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fy)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fy)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fy)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fy)})n)n)n)n ⊢ ∀_{obj}\underline{m}: \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}] ⊢ \overline{(s1)} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fx)}[\underline{m}]\}\} ⊢ eqReflexivity ≫ \underline{m} = \underline{m}; A4 @ \underline{m} ⊳ ∀_{obj}\underline{m}: \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}] ≫ \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}]; sameOrderedPair ⊳ \underline{m} = \underline{m} ⊳ \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}] ≫ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fx)}[\underline{m}]\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fy)}[\underline{m}]\}\}; eqTransitivity ⊳ \overline{(s1)} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fx)}[\underline{m}]\}\} ⊳ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fx)}[\underline{m}]\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fy)}[\underline{m}]\}\} ≫ \overline{(s1)} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fy)}[\underline{m}]\}\}; eqSymmetry ⊳ \overline{(s1)} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fy)}[\underline{m}]\}\} ≫ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fy)}[\underline{m}]\}\} = \overline{(s1)}; NatType ≫ \underline{m} ∈ N; MemberOfSeries ⊳ \underline{m} ∈ N ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fy)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fy)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fy)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fy)})n)n)n)n ≫ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fy)}[\underline{m}]\}\} ∈ \underline{(fy)}; SameMember ⊳ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fy)}[\underline{m}]\}\} = \overline{(s1)} ⊳ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fy)}[\underline{m}]\}\} ∈ \underline{(fy)} ≫ \overline{(s1)} ∈ \underline{(fy)}; ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: Ded ⊳ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fy)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fy)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fy)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fy)})n)n)n)n ⊢ ∀_{obj}\underline{m}: \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}] ⊢ \overline{(s1)} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fx)}[\underline{m}]\}\} ⊢ \overline{(s1)} ∈ \underline{(fy)} ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fy)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fy)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fy)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fy)})n)n)n)n ⇒ ∀_{obj}\underline{m}: \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}] ⇒ \overline{(s1)} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fx)}[\underline{m}]\}\} ⇒ \overline{(s1)} ∈ \underline{(fy)}⌉, p_{0}, c)]

[To = f(Subset)(Helper) →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fy)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fy)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fy)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fy)})n)n)n)n ⇒ ∀_{obj}\underline{m}: \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}] ⇒ \overline{(s1)} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fx)}[\underline{m}]\}\} ⇒ \overline{(s1)} ∈ \underline{(fy)}]

[To = f(Subset)(Helper) →^{tex} “To=f(Subset)(Helper)”] [To = f(Subset)(Helper) →^{pyk} “lemma to=f subset helper”]

To = f(Subset)

[To = f(Subset) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fy)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fy)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fy)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fy)})n)n)n)n ⊢ ∀_{obj}\underline{m}: \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}] ⊢ \overline{(s1)} ∈ \underline{(fx)} ⊢ InSeries ≫ ˙¬ (∀_{obj}\underline{m}: ˙¬ (\overline{(s1)} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fx)}[\underline{m}]\}\})n)n; To = f(Subset)(Helper) ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fy)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fy)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fy)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fy)})n)n)n)n ⇒ ∀_{obj}\underline{m}: \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}] ⇒ \overline{(s1)} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fx)}[\underline{m}]\}\} ⇒ \overline{(s1)} ∈ \underline{(fy)}; MP2 ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fy)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fy)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fy)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fy)})n)n)n)n ⇒ ∀_{obj}\underline{m}: \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}] ⇒ \overline{(s1)} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fx)}[\underline{m}]\}\} ⇒ \overline{(s1)} ∈ \underline{(fy)} ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fy)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fy)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fy)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fy)})n)n)n)n ⊳ ∀_{obj}\underline{m}: \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}] ≫ \overline{(s1)} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fx)}[\underline{m}]\}\} ⇒ \overline{(s1)} ∈ \underline{(fy)}; ExistMP ⊳ \overline{(s1)} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fx)}[\underline{m}]\}\} ⇒ \overline{(s1)} ∈ \underline{(fy)} ⊳ ˙¬ (∀_{obj}\underline{m}: ˙¬ (\overline{(s1)} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fx)}[\underline{m}]\}\})n)n ≫ \overline{(s1)} ∈ \underline{(fy)}; ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: Ded ⊳ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fy)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fy)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fy)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fy)})n)n)n)n ⊢ ∀_{obj}\underline{m}: \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}] ⊢ \overline{(s1)} ∈ \underline{(fx)} ⊢ \overline{(s1)} ∈ \underline{(fy)} ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fy)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fy)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fy)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fy)})n)n)n)n ⇒ ∀_{obj}\underline{m}: \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}] ⇒ \overline{(s1)} ∈ \underline{(fx)} ⇒ \overline{(s1)} ∈ \underline{(fy)}; λc.Typeseries0(⌈\underline{(fx)}⌉, ⌈Q⌉) ⊢⊢ λc.Typeseries0(⌈\underline{(fy)}⌉, ⌈Q⌉) ⊢⊢ ∀_{obj}\underline{m}: \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}] ⊢ SeriesType ⊳ ⊳ λc.Typeseries0(⌈\underline{(fx)}⌉, ⌈Q⌉) ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n; SeriesType ⊳ ⊳ λc.Typeseries0(⌈\underline{(fy)}⌉, ⌈Q⌉) ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fy)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fy)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fy)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fy)})n)n)n)n; MP3 ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ⇒ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fy)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fy)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fy)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fy)})n)n)n)n ⇒ ∀_{obj}\underline{m}: \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}] ⇒ \overline{(s1)} ∈ \underline{(fx)} ⇒ \overline{(s1)} ∈ \underline{(fy)} ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fy)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fy)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fy)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fy)})n)n)n)n⊳∀_{obj}\underline{m}: \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}] ≫ \overline{(s1)} ∈ \underline{(fx)} ⇒ \overline{(s1)} ∈ \underline{(fy)}; Gen⊳\overline{(s1)} ∈ \underline{(fx)} ⇒ \overline{(s1)} ∈ \underline{(fy)} ≫ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \underline{(fx)} ⇒ \overline{(s1)} ∈ \underline{(fy)}; Repetition ⊳ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \underline{(fx)} ⇒ \overline{(s1)} ∈ \underline{(fy)} ≫ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \underline{(fx)} ⇒ \overline{(s1)} ∈ \underline{(fy)}⌉, p_{0}, c)]

[To = f(Subset) →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: λc.Typeseries0(⌈\underline{(fx)}⌉, ⌈Q⌉) ⊢⊢ λc.Typeseries0(⌈\underline{(fy)}⌉, ⌈Q⌉) ⊢⊢ ∀_{obj}\underline{m}: \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}] ⊢ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \underline{(fx)} ⇒ \overline{(s1)} ∈ \underline{(fy)}]

[To = f(Subset) →^{tex} “To=f(Subset)”]

[To = f(Subset) →^{pyk} “lemma to=f subset”]

To = f

[To = f →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: λc.Typeseries0(⌈\underline{(fx)}⌉, ⌈Q⌉) ⊢⊢ λc.Typeseries0(⌈\underline{(fy)}⌉, ⌈Q⌉) ⊢⊢ ∀_{obj}\underline{m}: \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}] ⊢ A4 @ \underline{m} ⊳ ∀_{obj}\underline{m}: \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}] ≫ \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}]; eqSymmetry ⊳ \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}] ≫ \underline{(fy)}[\underline{m}] = \underline{(fx)}[\underline{m}]; Gen ⊳ \underline{(fy)}[\underline{m}] = \underline{(fx)}[\underline{m}] ≫ ∀_{obj}\underline{m}: \underline{(fy)}[\underline{m}] = \underline{(fx)}[\underline{m}]; To = f(Subset) ⊳ ⊳ λc.Typeseries0(⌈\underline{(fx)}⌉, ⌈Q⌉) ⊳ ⊳ λc.Typeseries0(⌈\underline{(fy)}⌉, ⌈Q⌉) ⊳ ∀_{obj}\underline{m}: \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}] ≫ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \underline{(fx)} ⇒ \overline{(s1)} ∈ \underline{(fy)}; To = f(Subset) ⊳ ⊳ λc.Typeseries0(⌈\underline{(fy)}⌉, ⌈Q⌉) ⊳ ⊳ λc.Typeseries0(⌈\underline{(fx)}⌉, ⌈Q⌉) ⊳ ∀_{obj}\underline{m}: \underline{(fy)}[\underline{m}] = \underline{(fx)}[\underline{m}] ≫ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \underline{(fy)} ⇒ \overline{(s1)} ∈ \underline{(fx)}; ToSetEquality ⊳ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \underline{(fx)} ⇒ \overline{(s1)} ∈ \underline{(fy)} ⊳ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ \underline{(fy)} ⇒ \overline{(s1)} ∈ \underline{(fx)} ≫ \underline{(fx)} = \underline{(fy)}⌉, p_{0}, c)]

[To = f →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: λc.Typeseries0(⌈\underline{(fx)}⌉, ⌈Q⌉) ⊢⊢ λc.Typeseries0(⌈\underline{(fy)}⌉, ⌈Q⌉) ⊢⊢ ∀_{obj}\underline{m}: \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}] ⊢ \underline{(fx)} = \underline{(fy)}]

[To = f →^{tex} “To=f”]

[To = f →^{pyk} “lemma to=f”]

productIsFunction

[productIsFunction →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(m1)}: ∀\underline{(m2)}: ∀\underline{(fx)}: ∀\underline{(fy)}: \overline{(f1)} = \overline{(f3)} ⊢ \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{(m1)}, \underline{(m1)}\}, \{\underline{(m1)}, (\underline{(fx)}[\underline{(m1)}] ∗ \underline{(fy)}[\underline{(m1)}])\}\} ⊢ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{(m2)}, \underline{(m2)}\}, \{\underline{(m2)}, (\underline{(fx)}[\underline{(m2)}] ∗ \underline{(fy)}[\underline{(m2)}])\}\} ⊢ FromOrderedPair(1) ⊳ \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{(m1)}, \underline{(m1)}\}, \{\underline{(m1)}, (\underline{(fx)}[\underline{(m1)}] ∗ \underline{(fy)}[\underline{(m1)}])\}\} ≫ \overline{(f1)} = \underline{(m1)}; eqSymmetry ⊳ \overline{(f1)} = \underline{(m1)} ≫ \underline{(m1)} = \overline{(f1)}; FromOrderedPair(1) ⊳ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{(m2)}, \underline{(m2)}\}, \{\underline{(m2)}, (\underline{(fx)}[\underline{(m2)}] ∗ \underline{(fy)}[\underline{(m2)}])\}\} ≫ \overline{(f3)} = \underline{(m2)}; eqTransitivity4 ⊳ \underline{(m1)} = \overline{(f1)} ⊳ \overline{(f1)} = \overline{(f3)} ⊳ \overline{(f3)} = \underline{(m2)} ≫ \underline{(m1)} = \underline{(m2)}; SameSeries⊳\underline{(m1)} = \underline{(m2)} ≫ \underline{(fx)}[\underline{(m1)}] = \underline{(fx)}[\underline{(m2)}]; SameSeries⊳\underline{(m1)} = \underline{(m2)} ≫ \underline{(fy)}[\underline{(m1)}] = \underline{(fy)}[\underline{(m2)}]; MultiplyEquations ⊳ \underline{(fx)}[\underline{(m1)}] = \underline{(fx)}[\underline{(m2)}] ⊳ \underline{(fy)}[\underline{(m1)}] = \underline{(fy)}[\underline{(m2)}] ≫ (\underline{(fx)}[\underline{(m1)}] ∗ \underline{(fy)}[\underline{(m1)}]) = (\underline{(fx)}[\underline{(m2)}] ∗ \underline{(fy)}[\underline{(m2)}]); FromOrderedPair(2) ⊳ \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{(m1)}, \underline{(m1)}\}, \{\underline{(m1)}, (\underline{(fx)}[\underline{(m1)}] ∗ \underline{(fy)}[\underline{(m1)}])\}\} ≫ \overline{(f2)} = (\underline{(fx)}[\underline{(m1)}] ∗ \underline{(fy)}[\underline{(m1)}]); FromOrderedPair(2) ⊳ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{(m2)}, \underline{(m2)}\}, \{\underline{(m2)}, (\underline{(fx)}[\underline{(m2)}] ∗ \underline{(fy)}[\underline{(m2)}])\}\} ≫ \overline{(f4)} = (\underline{(fx)}[\underline{(m2)}] ∗ \underline{(fy)}[\underline{(m2)}]); eqSymmetry ⊳ \overline{(f4)} = (\underline{(fx)}[\underline{(m2)}] ∗ \underline{(fy)}[\underline{(m2)}]) ≫ (\underline{(fx)}[\underline{(m2)}] ∗ \underline{(fy)}[\underline{(m2)}]) = \overline{(f4)}; eqTransitivity4 ⊳ \overline{(f2)} = (\underline{(fx)}[\underline{(m1)}] ∗ \underline{(fy)}[\underline{(m1)}]) ⊳ (\underline{(fx)}[\underline{(m1)}] ∗ \underline{(fy)}[\underline{(m1)}]) = (\underline{(fx)}[\underline{(m2)}] ∗ \underline{(fy)}[\underline{(m2)}]) ⊳ (\underline{(fx)}[\underline{(m2)}] ∗ \underline{(fy)}[\underline{(m2)}]) = \overline{(f4)} ≫ \overline{(f2)} = \overline{(f4)}; ∀\underline{(m1)}: ∀\underline{(m2)}: ∀\underline{(fx)}: ∀\underline{(fy)}: Ded ⊳ ∀\underline{(m1)}: ∀\underline{(m2)}: ∀\underline{(fx)}: ∀\underline{(fy)}: \overline{(f1)} = \overline{(f3)} ⊢ \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} = \{\{\underline{(m1)}, \underline{(m1)}\}, \{\underline{(m1)}, (\underline{(fx)}[\underline{(m1)}] ∗ \underline{(fy)}[\underline{(m1)}])\}\} ⊢ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} = \{\{\underline{(m2)}, \underline{(m2)}\}, \{\underline{(m2)}, (\underline{(fx)}[\underline{(m2)}] ∗ \underline{(fy)}[\underline{(m2)}])\}\} ⊢ \overline{(f2)} = \overline{(f4)} ≫ ∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ {ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)}⌉, p_{0}, c)]

[productIsFunction →^{stmt} SystemQ ⊢ ∀\underline{(m1)}: ∀\underline{(m2)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ {ph ∈ {ph ∈ P(P(Union({N, Q}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)}]

[productIsFunction →^{tex} “productIsFunction”]

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

productIsTotal

[productIsTotal →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: \overline{(s1)} ∈ N ⊢ SeriesType ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n; SeriesType ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fy)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fy)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fy)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fy)})n)n)n)n; ValueType ⊳ \overline{(s1)} ∈ N ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ≫ \underline{(fx)}[\overline{(s1)}] ∈ Q; ValueType ⊳ \overline{(s1)} ∈ N ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fy)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fy)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fy)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fy)})n)n)n)n ≫ \underline{(fy)}[\overline{(s1)}] ∈ Q; Qclosed(Multiplication) ⊳ \underline{(fx)}[\overline{(s1)}] ∈ Q ⊳ \underline{(fy)}[\overline{(s1)}] ∈ Q ≫ (\underline{(fx)}[\overline{(s1)}] ∗ \underline{(fy)}[\overline{(s1)}]) ∈ Q; ToCartProd ⊳ \overline{(s1)} ∈ N ⊳ (\underline{(fx)}[\overline{(s1)}] ∗ \underline{(fy)}[\overline{(s1)}]) ∈ Q ≫ \{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, (\underline{(fx)}[\overline{(s1)}] ∗ \underline{(fy)}[\overline{(s1)}])\}\} ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}; eqReflexivity ≫ \{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, (\underline{(fx)}[\overline{(s1)}] ∗ \underline{(fy)}[\overline{(s1)}])\}\} = \{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, (\underline{(fx)}[\overline{(s1)}] ∗ \underline{(fy)}[\overline{(s1)}])\}\}; IntroExist @ \overline{(s1)} ⊳ \{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, (\underline{(fx)}[\overline{(s1)}] ∗ \underline{(fy)}[\overline{(s1)}])\}\} = \{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, (\underline{(fx)}[\overline{(s1)}] ∗ \underline{(fy)}[\overline{(s1)}])\}\} ≫ ˙¬ (∀_{obj}\underline{m}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, (\underline{(fx)}[\overline{(s1)}] ∗ \underline{(fy)}[\overline{(s1)}])\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\})n)n; Formula2Sep ⊳ \{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, (\underline{(fx)}[\overline{(s1)}] ∗ \underline{(fy)}[\overline{(s1)}])\}\} ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ⊳ ˙¬ (∀_{obj}\underline{m}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, (\underline{(fx)}[\overline{(s1)}] ∗ \underline{(fy)}[\overline{(s1)}])\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\})n)n ≫ \{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, (\underline{(fx)}[\overline{(s1)}] ∗ \underline{(fy)}[\overline{(s1)}])\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}; Repetition ⊳ \{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, (\underline{(fx)}[\overline{(s1)}] ∗ \underline{(fy)}[\overline{(s1)}])\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\} ≫ \{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, (\underline{(fx)}[\overline{(s1)}] ∗ \underline{(fy)}[\overline{(s1)}])\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\}; IntroExist @(\underline{(fx)}[\overline{(s1)}] ∗ \underline{(fy)}[\overline{(s1)}]) ⊳ \{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, (\underline{(fx)}[\overline{(s1)}] ∗ \underline{(fy)}[\overline{(s1)}])\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\} ≫ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\})n)n; ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: Ded ⊳ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: \overline{(s1)} ∈ N ⊢ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\})n)n ≫ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\})n)n⌉, p_{0}, c)]

[productIsTotal →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\})n)n]

[productIsTotal →^{tex} “productIsTotal”]

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

ProductIsRationalSeries

[ProductIsRationalSeries →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{(m1)}: ∀\underline{(m2)}: ∀\underline{(fx)}: ∀\underline{(fy)}: CPseparationIsRelation ≫ ∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n; productIsFunction ≫ ∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)}; productIsTotal ≫ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\})n)n; ToSeries ⊳ ∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⊳ ∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)} ⊳ ∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\})n)n ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\})n)n)n)n⌉, p_{0}, c)]

[ProductIsRationalSeries →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{(m1)}: ∀\underline{(m2)}: ∀\underline{(fx)}: ∀\underline{(fy)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\})n)n)n)n]

[ProductIsRationalSeries →^{tex} “ProductIsRationalSeries”]

[ProductIsRationalSeries →^{pyk} “lemma productIsRationalSeries”]

TimesF

[TimesF →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: λc.TypeNat0(⌈\underline{m}⌉) ⊢⊢ λc.Typeseries0(⌈\underline{(fx)}⌉, ⌈Q⌉) ⊢⊢ λc.Typeseries0(⌈\underline{(fy)}⌉, ⌈Q⌉) ⊢⊢ NatType ≫ \underline{m} ∈ N; SeriesType ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n; SeriesType ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fy)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fy)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fy)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fy)})n)n)n)n; ProductIsRationalSeries ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\})n)n)n)n; MemberOfSeries ⊳ \underline{m} ∈ N ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\})n)n)n)n ≫ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}])\}\})n)n\}; ValueType⊳\underline{m} ∈ N⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ≫ \underline{(fx)}[\underline{m}] ∈ Q; ValueType ⊳ \underline{m} ∈ N ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fy)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fy)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fy)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fy)})n)n)n)n ≫ \underline{(fy)}[\underline{m}] ∈ Q; Qclosed(Multiplication) ⊳ \underline{(fx)}[\underline{m}] ∈ Q ⊳ \underline{(fy)}[\underline{m}] ∈ Q ≫ (\underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}]) ∈ Q; ToCartProd ⊳ \underline{m} ∈ N ⊳ (\underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}]) ∈ Q ≫ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\} ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\}; eqReflexivity ≫ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\}; IntroExist @ \underline{m} ⊳ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\} ≫ ˙¬ (∀_{obj}\underline{m}: ˙¬ (\{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\})n)n; Formula2Sep ⊳ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\} ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n\} ⊳ ˙¬ (∀_{obj}\underline{m}: ˙¬ (\{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\})n)n ≫ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\})n)n\}; eqReflexivity ≫ \underline{m} = \underline{m}; UniqueMember ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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\})n)n)n)n ⊳ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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{(fy)}[\underline{m}])\}\})n)n\} ⊳ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\})n)n\} ⊳ \underline{m} = \underline{m} ≫ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] = (\underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])⌉, p_{0}, c)]

[TimesF →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: λc.TypeNat0(⌈\underline{m}⌉) ⊢⊢ λc.Typeseries0(⌈\underline{(fx)}⌉, ⌈Q⌉) ⊢⊢ λc.Typeseries0(⌈\underline{(fy)}⌉, ⌈Q⌉) ⊢⊢ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] = (\underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])]

[TimesF →^{tex} “TimesF”]

[TimesF →^{pyk} “lemma timesF”]

−x + (1/2)x = −(1/2)x

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

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

PositiveTripled

[PositiveTripled →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{x}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{x}))n)n)n ⊢ 0 < 3 ≫ ˙¬ (0 <= ((1 + 1) + 1) ⇒ ˙¬ ( ˙¬ (0 = ((1 + 1) + 1))n)n)n; PositiveFactors ⊳ ˙¬ (0 <= ((1 + 1) + 1) ⇒ ˙¬ ( ˙¬ (0 = ((1 + 1) + 1))n)n)n ⊳ ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{x}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{x}))n)n)n ≫ ˙¬ (0 <= (((1 + 1) + 1) ∗ (rec((1 + 1) + 1) ∗ \underline{x})) ⇒ ˙¬ ( ˙¬ (0 = (((1 + 1) + 1) ∗ (rec((1 + 1) + 1) ∗ \underline{x})))n)n)n; timesAssociativity ≫ ((((1 + 1) + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{x}) = (((1 + 1) + 1) ∗ (rec((1 + 1) + 1) ∗ \underline{x})); eqSymmetry ⊳ ((((1 + 1) + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{x}) = (((1 + 1) + 1) ∗ (rec((1 + 1) + 1) ∗ \underline{x})) ≫ (((1 + 1) + 1) ∗ (rec((1 + 1) + 1) ∗ \underline{x})) = ((((1 + 1) + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{x}); PositiveNonzero ⊳ ˙¬ (0 <= ((1 + 1) + 1) ⇒ ˙¬ ( ˙¬ (0 = ((1 + 1) + 1))n)n)n ≫ ˙¬ (((1 + 1) + 1) = 0)n; Reciprocal ⊳ ˙¬ (((1 + 1) + 1) = 0)n ≫ (((1 + 1) + 1) ∗ rec((1 + 1) + 1)) = 1; eqMultiplication ⊳ (((1 + 1) + 1) ∗ rec((1 + 1) + 1)) = 1 ≫ ((((1 + 1) + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{x}) = (1 ∗ \underline{x}); times1Left ≫ (1 ∗ \underline{x}) = \underline{x}; eqTransitivity4 ⊳ (((1 + 1) + 1) ∗ (rec((1 + 1) + 1) ∗ \underline{x})) = ((((1 + 1) + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{x}) ⊳ ((((1 + 1) + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{x}) = (1 ∗ \underline{x}) ⊳ (1 ∗ \underline{x}) = \underline{x} ≫ (((1 + 1) + 1) ∗ (rec((1 + 1) + 1) ∗ \underline{x})) = \underline{x}; SubLessRight ⊳ (((1 + 1) + 1) ∗ (rec((1 + 1) + 1) ∗ \underline{x})) = \underline{x} ⊳ ˙¬ (0 <= (((1 + 1) + 1) ∗ (rec((1 + 1) + 1) ∗ \underline{x})) ⇒ ˙¬ ( ˙¬ (0 = (((1 + 1) + 1) ∗ (rec((1 + 1) + 1) ∗ \underline{x})))n)n)n ≫ ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n⌉, p_{0}, c)]

[PositiveTripled →^{stmt} SystemQ ⊢ ∀\underline{x}: ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{x}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{x}))n)n)n ⊢ ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n]

[PositiveTripled →^{tex} “PositiveTripled”]

[PositiveTripled →^{pyk} “lemma positiveTripled”]

PositiveDividedBy3

[PositiveDividedBy3 →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ⊢ 0 < 3 ≫ ˙¬ (0 <= ((1 + 1) + 1) ⇒ ˙¬ ( ˙¬ (0 = ((1 + 1) + 1))n)n)n; PositiveInverted ⊳ ˙¬ (0 <= ((1 + 1) + 1) ⇒ ˙¬ ( ˙¬ (0 = ((1 + 1) + 1))n)n)n ≫ ˙¬ (0 <= rec((1 + 1) + 1) ⇒ ˙¬ ( ˙¬ (0 = rec((1 + 1) + 1))n)n)n; PositiveFactors ⊳ ˙¬ (0 <= rec((1 + 1) + 1) ⇒ ˙¬ ( ˙¬ (0 = rec((1 + 1) + 1))n)n)n ⊳ ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ≫ ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{x}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{x}))n)n)n⌉, p_{0}, c)]

[PositiveDividedBy3 →^{stmt} SystemQ ⊢ ∀\underline{x}: ˙¬ (0 <= \underline{x} ⇒ ˙¬ ( ˙¬ (0 = \underline{x})n)n)n ⊢ ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{x}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{x}))n)n)n]

[PositiveDividedBy3 →^{tex} “PositiveDividedBy3”]

[PositiveDividedBy3 →^{pyk} “lemma positiveDividedBy3”]

|x − x| = 0

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

[|x − x| = 0 →^{stmt} SystemQ ⊢ ∀\underline{x}: |(\underline{x} + (−u\underline{x}))| = 0]

[|x − x| = 0 →^{tex} “|x-x|=0”]

[|x − x| = 0 →^{pyk} “lemma |x-x|=0”]

1 < 2

[1 < 2 →^{proof} λc.λx.P(⌈SystemQ ⊢ 0 < 1 ≫ ˙¬ (0 <= 1 ⇒ ˙¬ ( ˙¬ (0 = 1)n)n)n; LessAddition ⊳ ˙¬ (0 <= 1 ⇒ ˙¬ ( ˙¬ (0 = 1)n)n)n ≫ ˙¬ ((0 + 1) <= (1 + 1) ⇒ ˙¬ ( ˙¬ ((0 + 1) = (1 + 1))n)n)n; plus0Left ≫ (0 + 1) = 1; SubLessLeft ⊳ (0 + 1) = 1 ⊳ ˙¬ ((0 + 1) <= (1 + 1) ⇒ ˙¬ ( ˙¬ ((0 + 1) = (1 + 1))n)n)n ≫ ˙¬ (1 <= (1 + 1) ⇒ ˙¬ ( ˙¬ (1 = (1 + 1))n)n)n; Repetition ⊳ ˙¬ (1 <= (1 + 1) ⇒ ˙¬ ( ˙¬ (1 = (1 + 1))n)n)n ≫ ˙¬ (1 <= (1 + 1) ⇒ ˙¬ ( ˙¬ (1 = (1 + 1))n)n)n⌉, p_{0}, c)]

[1 < 2 →^{stmt} SystemQ ⊢ ˙¬ (1 <= (1 + 1) ⇒ ˙¬ ( ˙¬ (1 = (1 + 1))n)n)n]

[1 < 2 →^{tex} “1<2”]

[1 < 2 →^{pyk} “lemma 1<2”]

1/3 < 2/3

[1/3 < 2/3 →^{proof} λc.λx.P(⌈SystemQ ⊢ 1 < 2 ≫ ˙¬ (1 <= (1 + 1) ⇒ ˙¬ ( ˙¬ (1 = (1 + 1))n)n)n; 0 < 1/3 ≫ ˙¬ (0 <= rec((1 + 1) + 1) ⇒ ˙¬ ( ˙¬ (0 = rec((1 + 1) + 1))n)n)n; LessMultiplication ⊳ ˙¬ (0 <= rec((1 + 1) + 1) ⇒ ˙¬ ( ˙¬ (0 = rec((1 + 1) + 1))n)n)n ⊳ ˙¬ (1 <= (1 + 1) ⇒ ˙¬ ( ˙¬ (1 = (1 + 1))n)n)n ≫ ˙¬ ((1 ∗ rec((1 + 1) + 1)) <= ((1 + 1) ∗ rec((1 + 1) + 1)) ⇒ ˙¬ ( ˙¬ ((1 ∗ rec((1 + 1) + 1)) = ((1 + 1) ∗ rec((1 + 1) + 1)))n)n)n; times1Left ≫ (1 ∗ rec((1 + 1) + 1)) = rec((1 + 1) + 1); SubLessLeft ⊳ (1 ∗ rec((1 + 1) + 1)) = rec((1 + 1) + 1) ⊳ ˙¬ ((1 ∗ rec((1 + 1) + 1)) <= ((1 + 1) ∗ rec((1 + 1) + 1)) ⇒ ˙¬ ( ˙¬ ((1 ∗ rec((1 + 1) + 1)) = ((1 + 1) ∗ rec((1 + 1) + 1)))n)n)n ≫ ˙¬ (rec((1 + 1) + 1) <= ((1 + 1) ∗ rec((1 + 1) + 1)) ⇒ ˙¬ ( ˙¬ (rec((1 + 1) + 1) = ((1 + 1) ∗ rec((1 + 1) + 1)))n)n)n; Repetition ⊳ ˙¬ (rec((1 + 1) + 1) <= ((1 + 1) ∗ rec((1 + 1) + 1)) ⇒ ˙¬ ( ˙¬ (rec((1 + 1) + 1) = ((1+1)∗rec((1+1)+1)))n)n)n ≫ ˙¬ (rec((1+1)+1) <= ((1+1)∗rec((1+1)+1)) ⇒ ˙¬ ( ˙¬ (rec((1 + 1) + 1) = ((1 + 1) ∗ rec((1 + 1) + 1)))n)n)n⌉, p_{0}, c)]

[1/3 < 2/3 →^{stmt} SystemQ ⊢ ˙¬ (rec((1 + 1) + 1) <= ((1 + 1) ∗ rec((1 + 1) + 1)) ⇒ ˙¬ ( ˙¬ (rec((1 + 1) + 1) = ((1 + 1) ∗ rec((1 + 1) + 1)))n)n)n]

[1/3 < 2/3 →^{tex} “1/3<2/3”]

[1/3 < 2/3 →^{pyk} “lemma 1/3<2/3”]

(1/3)x + (1/3)x = (2/3)x

[(1/3)x + (1/3)x = (2/3)x →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: TwoWholes ≫ ((rec((1 + 1) + 1) ∗ \underline{x}) + (rec((1 + 1) + 1) ∗ \underline{x})) = ((1+1)∗(rec((1+1)+1)∗\underline{x})); timesAssociativity ≫ (((1+1)∗rec((1+1)+1))∗\underline{x}) = ((1 + 1) ∗ (rec((1 + 1) + 1) ∗ \underline{x})); eqSymmetry ⊳ (((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{x}) = ((1 + 1) ∗ (rec((1 + 1) + 1) ∗ \underline{x})) ≫ ((1 + 1) ∗ (rec((1 + 1) + 1) ∗ \underline{x})) = (((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{x}); eqTransitivity ⊳ ((rec((1 + 1) + 1) ∗ \underline{x}) + (rec((1 + 1) + 1) ∗ \underline{x})) = ((1 + 1) ∗ (rec((1 + 1) + 1) ∗ \underline{x})) ⊳ ((1 + 1) ∗ (rec((1 + 1) + 1) ∗ \underline{x})) = (((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{x}) ≫ ((rec((1 + 1) + 1) ∗ \underline{x}) + (rec((1 + 1) + 1) ∗ \underline{x})) = (((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{x}); Repetition ⊳ ((rec((1 + 1) + 1) ∗ \underline{x}) + (rec((1 + 1) + 1) ∗ \underline{x})) = (((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{x}) ≫ ((rec((1+1)+1)∗\underline{x})+(rec((1+1)+1)∗\underline{x})) = (((1+1)∗rec((1+1)+1))∗\underline{x})⌉, p_{0}, c)]

[(1/3)x + (1/3)x = (2/3)x →^{stmt} SystemQ ⊢ ∀\underline{x}: ((rec((1 + 1) + 1) ∗ \underline{x}) + (rec((1 + 1) + 1) ∗ \underline{x})) = (((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{x})]

[(1/3)x + (1/3)x = (2/3)x →^{tex} “(1/3)x+(1/3)x=(2/3)x”]

[(1/3)x + (1/3)x = (2/3)x →^{pyk} “lemma (1/3)x+(1/3)x=(2/3)x”]

(2/3)x + (1/3)x = x

[(2/3)x + (1/3)x = x →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: TwoWholes ≫ ((rec((1 + 1) + 1) ∗ \underline{x}) + (rec((1 + 1) + 1) ∗ \underline{x})) = ((1+1)∗(rec((1+1)+1)∗\underline{x})); timesAssociativity ≫ (((1+1)∗rec((1+1)+1))∗\underline{x}) = ((1 + 1) ∗ (rec((1 + 1) + 1) ∗ \underline{x})); eqSymmetry ⊳ (((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{x}) = ((1 + 1) ∗ (rec((1 + 1) + 1) ∗ \underline{x})) ≫ ((1 + 1) ∗ (rec((1 + 1) + 1) ∗ \underline{x})) = (((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{x}); eqTransitivity ⊳ ((rec((1 + 1) + 1) ∗ \underline{x}) + (rec((1 + 1) + 1) ∗ \underline{x})) = ((1 + 1) ∗ (rec((1 + 1) + 1) ∗ \underline{x})) ⊳ ((1 + 1) ∗ (rec((1 + 1) + 1) ∗ \underline{x})) = (((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{x}) ≫ ((rec((1 + 1) + 1) ∗ \underline{x}) + (rec((1 + 1) + 1) ∗ \underline{x})) = (((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{x}); eqAddition ⊳ ((rec((1 + 1) + 1) ∗ \underline{x}) + (rec((1 + 1) + 1) ∗ \underline{x})) = (((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{x}) ≫ (((rec((1 + 1) + 1) ∗ \underline{x}) + (rec((1 + 1) + 1) ∗ \underline{x})) + (rec((1 + 1) + 1) ∗ \underline{x})) = ((((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{x}) + (rec((1 + 1) + 1) ∗ \underline{x})); ThreeThirds ≫ (((rec((1 + 1) + 1) ∗ \underline{x}) + (rec((1 + 1) + 1) ∗ \underline{x})) + (rec((1 + 1) + 1) ∗ \underline{x})) = \underline{x}; Equality⊳(((rec((1+1)+1)∗\underline{x})+(rec((1+1)+1)∗\underline{x}))+(rec((1+1)+1)∗\underline{x})) = ((((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{x}) + (rec((1 + 1) + 1) ∗ \underline{x})) ⊳ (((rec((1 + 1) + 1) ∗ \underline{x}) + (rec((1 + 1) + 1) ∗ \underline{x})) + (rec((1 + 1) + 1) ∗ \underline{x})) = \underline{x} ≫ ((((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{x}) + (rec((1 + 1) + 1) ∗ \underline{x})) = \underline{x}⌉, p_{0}, c)]

[(2/3)x + (1/3)x = x →^{stmt} SystemQ ⊢ ∀\underline{x}: ((((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{x}) + (rec((1 + 1) + 1) ∗ \underline{x})) = \underline{x}]

[(2/3)x + (1/3)x = x →^{tex} “(2/3)x+(1/3)x=x”]

[(2/3)x + (1/3)x = x →^{pyk} “lemma (2/3)x+(1/3)x=x”]

−x + (2/3)x = −(1/3)x

[−x + (2/3)x = −(1/3)x →^{tex} “-x+(2/3)x=-(1/3)x”]

[−x + (2/3)x = −(1/3)x →^{pyk} “lemma -x+(2/3)x=-(1/3)x”]

−(1/3)x − (1/3)x = −(2/3)x

[−(1/3)x − (1/3)x = −(2/3)x →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: (1/3)x + (1/3)x = (2/3)x ≫ ((rec((1 + 1) + 1) ∗ \underline{x}) + (rec((1 + 1) + 1) ∗ \underline{x})) = (((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{x}); EqNegated ⊳ ((rec((1 + 1) + 1) ∗ \underline{x}) + (rec((1 + 1) + 1) ∗ \underline{x})) = (((1+1)∗rec((1+1)+1))∗\underline{x}) ≫ (−u((rec((1+1)+1)∗\underline{x})+(rec((1+1)+1)∗\underline{x}))) = (−u(((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{x})); −x − y = −(x + y) ≫ ((−u(rec((1 + 1) + 1) ∗ \underline{x})) + (−u(rec((1 + 1) + 1) ∗ \underline{x}))) = (−u((rec((1 + 1) + 1) ∗ \underline{x}) + (rec((1 + 1) + 1) ∗ \underline{x}))); eqTransitivity ⊳ ((−u(rec((1 + 1) + 1) ∗ \underline{x})) + (−u(rec((1 + 1) + 1) ∗ \underline{x}))) = (−u((rec((1 + 1) + 1) ∗ \underline{x}) + (rec((1 + 1) + 1) ∗ \underline{x}))) ⊳ (−u((rec((1 + 1) + 1) ∗ \underline{x}) + (rec((1+1)+1)∗\underline{x}))) = (−u(((1+1)∗rec((1+1)+1))∗\underline{x})) ≫ ((−u(rec((1+1)+ 1) ∗ \underline{x})) + (−u(rec((1 + 1) + 1) ∗ \underline{x}))) = (−u(((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{x}))⌉, p_{0}, c)]

[−(1/3)x − (1/3)x = −(2/3)x →^{stmt} SystemQ ⊢ ∀\underline{x}: ((−u(rec((1 + 1) + 1) ∗ \underline{x})) + (−u(rec((1 + 1) + 1) ∗ \underline{x}))) = (−u(((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{x}))]

[−(1/3)x − (1/3)x = −(2/3)x →^{tex} “-(1/3)x-(1/3)x=-(2/3)x”]

[−(1/3)x − (1/3)x = −(2/3)x →^{pyk} “lemma -(1/3)x-(1/3)x=-(2/3)x”]

−x + (1/3)x = −(2/3)x

[−x + (1/3)x = −(2/3)x →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: (2/3)x + (1/3)x = x ≫ ((((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{x}) + (rec((1 + 1) + 1) ∗ \underline{x})) = \underline{x}; PositiveToRight(Eq)⊳((((1+1)∗rec((1+1)+1))∗\underline{x})+(rec((1+1)+1)∗\underline{x})) = \underline{x} ≫ (((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{x}) = (\underline{x} + (−u(rec((1 + 1) + 1) ∗ \underline{x}))); EqNegated ⊳ (((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{x}) = (\underline{x} + (−u(rec((1 + 1) + 1) ∗ \underline{x}))) ≫ (−u(((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{x})) = (−u(\underline{x} + (−u(rec((1 + 1) + 1) ∗ \underline{x})))); MinusNegated ≫ (−u(\underline{x} + (−u(rec((1 + 1) + 1) ∗ \underline{x})))) = ((rec((1 + 1) + 1) ∗ \underline{x}) + (−u\underline{x})); plusCommutativity ≫ ((rec((1 + 1) + 1) ∗ \underline{x}) + (−u\underline{x})) = ((−u\underline{x})+(rec((1+1)+1)∗\underline{x})); eqTransitivity4⊳(−u(((1+1)∗rec((1+1)+1))∗\underline{x})) = (−u(\underline{x} + (−u(rec((1 + 1) + 1) ∗ \underline{x})))) ⊳ (−u(\underline{x} + (−u(rec((1 + 1) + 1) ∗ \underline{x})))) = ((rec((1 + 1) + 1) ∗ \underline{x}) + (−u\underline{x})) ⊳ ((rec((1 + 1) + 1) ∗ \underline{x}) + (−u\underline{x})) = ((−u\underline{x}) + (rec((1 + 1) + 1) ∗ \underline{x})) ≫ (−u(((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{x})) = ((−u\underline{x})+(rec((1+1)+1)∗\underline{x})); eqSymmetry⊳(−u(((1+1)∗rec((1+1)+1))∗\underline{x})) = ((−u\underline{x}) + (rec((1 + 1) + 1) ∗ \underline{x})) ≫ ((−u\underline{x}) + (rec((1 + 1) + 1) ∗ \underline{x})) = (−u(((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{x}))⌉, p_{0}, c)]

[−x + (1/3)x = −(2/3)x →^{stmt} SystemQ ⊢ ∀\underline{x}: ((−u\underline{x}) + (rec((1 + 1) + 1) ∗ \underline{x})) = (−u(((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{x}))]

[−x + (1/3)x = −(2/3)x →^{tex} “-x+(1/3)x=-(2/3)x”]

[−x + (1/3)x = −(2/3)x →^{pyk} “lemma -x+(1/3)x=-(2/3)x”]

PreserveLessGreater

[PreserveLessGreater →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(x1)}: ∀\underline{(x2)}: ∀\underline{(y1)}: ∀\underline{(y2)}: ∀\underline{z}: \underline{(x1)} <= (\underline{(y1)} + (−u\underline{z})) ⊢ ˙¬ (|(\underline{(x1)} + (−u\underline{(x2)}))| <= (rec((1 + 1) + 1) ∗ \underline{z}) ⇒ ˙¬ ( ˙¬ (|(\underline{(x1)} + (−u\underline{(x2)}))| = (rec((1 + 1) + 1) ∗ \underline{z}))n)n)n ⊢ ˙¬ (|(\underline{(y1)} + (−u\underline{(y2)}))| <= (rec((1 + 1) + 1) ∗ \underline{z}) ⇒ ˙¬ ( ˙¬ (|(\underline{(y1)} + (−u\underline{(y2)}))| = (rec((1 + 1) + 1) ∗ \underline{z}))n)n)n ⊢ 0 <= |x| ≫ 0 <= |(\underline{(x1)} + (−u\underline{(x2)}))|; leqLessTransitivity ⊳ 0 <= |(\underline{(x1)} + (−u\underline{(x2)}))| ⊳ ˙¬ (|(\underline{(x1)} + (−u\underline{(x2)}))| <= (rec((1 + 1) + 1) ∗ \underline{z}) ⇒ ˙¬ ( ˙¬ (|(\underline{(x1)} + (−u\underline{(x2)}))| = (rec((1 + 1) + 1) ∗ \underline{z}))n)n)n ≫ ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{z}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{z}))n)n)n; PositiveTripled ⊳ ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{z}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{z}))n)n)n ≫ ˙¬ (0 <= \underline{z} ⇒ ˙¬ ( ˙¬ (0 = \underline{z})n)n)n; NumericalDifferenceLess ⊳ ˙¬ (|(\underline{(x1)} + (−u\underline{(x2)}))| <= (rec((1 + 1) + 1) ∗ \underline{z}) ⇒ ˙¬ ( ˙¬ (|(\underline{(x1)} + (−u\underline{(x2)}))| = (rec((1 + 1) + 1) ∗ \underline{z}))n)n)n ≫ ˙¬ ( ˙¬ ((\underline{(x2)} + (−u(rec((1 + 1) + 1) ∗ \underline{z}))) <= \underline{(x1)} ⇒ ˙¬ ( ˙¬ ((\underline{(x2)} + (−u(rec((1 + 1) + 1) ∗ \underline{z}))) = \underline{(x1)})n)n)n ⇒ ˙¬ ( ˙¬ (\underline{(x1)} <= (\underline{(x2)} + (rec((1 + 1) + 1) ∗ \underline{z})) ⇒ ˙¬ ( ˙¬ (\underline{(x1)} = (\underline{(x2)}+(rec((1+1)+1)∗\underline{z})))n)n)n)n)n; FirstConjunct⊳ ˙¬ ( ˙¬ ((\underline{(x2)}+ (−u(rec((1 + 1) + 1) ∗ \underline{z}))) <= \underline{(x1)} ⇒ ˙¬ ( ˙¬ ((\underline{(x2)} + (−u(rec((1 + 1) + 1) ∗ \underline{z}))) = \underline{(x1)})n)n)n ⇒ ˙¬ ( ˙¬ (\underline{(x1)} <= (\underline{(x2)} + (rec((1 + 1) + 1) ∗ \underline{z})) ⇒ ˙¬ ( ˙¬ (\underline{(x1)} = (\underline{(x2)}+(rec((1+1)+1)∗\underline{z})))n)n)n)n)n ≫ ˙¬ ((\underline{(x2)}+(−u(rec((1+1)+1)∗\underline{z}))) <= \underline{(x1)} ⇒ ˙¬ ( ˙¬ ((\underline{(x2)} + (−u(rec((1 + 1) + 1) ∗ \underline{z}))) = \underline{(x1)})n)n)n; NegativeToRight(Less) ⊳ ˙¬ ((\underline{(x2)} + (−u(rec((1 + 1) + 1) ∗ \underline{z}))) <= \underline{(x1)} ⇒ ˙¬ ( ˙¬ ((\underline{(x2)} + (−u(rec((1 + 1) + 1) ∗ \underline{z}))) = \underline{(x1)})n)n)n ≫ ˙¬ (\underline{(x2)} <= (\underline{(x1)} + (rec((1 + 1) + 1) ∗ \underline{z})) ⇒ ˙¬ ( ˙¬ (\underline{(x2)} = (\underline{(x1)} + (rec((1 + 1) + 1) ∗ \underline{z})))n)n)n; leqAddition ⊳ \underline{(x1)} <= (\underline{(y1)} + (−u\underline{z})) ≫ (\underline{(x1)} + (rec((1 + 1) + 1) ∗ \underline{z})) <= ((\underline{(y1)} + (−u\underline{z})) + (rec((1 + 1) + 1) ∗ \underline{z})); −x + (1/3)x = −(2/3)x ≫ ((−u\underline{z}) + (rec((1 + 1) + 1) ∗ \underline{z})) = (−u(((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{z})); Three2twoTerms ⊳ ((−u\underline{z}) + (rec((1 + 1) + 1) ∗ \underline{z})) = (−u(((1 + 1) ∗ rec((1+1)+1))∗\underline{z})) ≫ ((\underline{(y1)}+(−u\underline{z}))+(rec((1+1)+1)∗\underline{z})) = (\underline{(y1)}+(−u(((1+ 1)∗rec((1+1)+1))∗\underline{z}))); subLeqRight⊳((\underline{(y1)}+(−u\underline{z}))+(rec((1+1)+1)∗\underline{z})) = (\underline{(y1)} + (−u(((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{z}))) ⊳ (\underline{(x1)} + (rec((1 + 1) + 1) ∗ \underline{z})) <= ((\underline{(y1)} + (−u\underline{z})) + (rec((1 + 1) + 1) ∗ \underline{z})) ≫ (\underline{(x1)} + (rec((1 + 1) + 1) ∗ \underline{z})) <= (\underline{(y1)} + (−u(((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{z}))); NumericalDifferenceLess ⊳ ˙¬ (|(\underline{(y1)} + (−u\underline{(y2)}))| <= (rec((1 + 1) + 1) ∗ \underline{z}) ⇒ ˙¬ ( ˙¬ (|(\underline{(y1)} + (−u\underline{(y2)}))| = (rec((1 + 1) + 1) ∗ \underline{z}))n)n)n ≫ ˙¬ ( ˙¬ ((\underline{(y2)} + (−u(rec((1 + 1) + 1) ∗ \underline{z}))) <= \underline{(y1)} ⇒ ˙¬ ( ˙¬ ((\underline{(y2)} + (−u(rec((1 + 1) + 1) ∗ \underline{z}))) = \underline{(y1)})n)n)n ⇒ ˙¬ ( ˙¬ (\underline{(y1)} <= (\underline{(y2)} + (rec((1 + 1) + 1) ∗ \underline{z})) ⇒ ˙¬ ( ˙¬ (\underline{(y1)} = (\underline{(y2)} + (rec((1 + 1) + 1) ∗ \underline{z})))n)n)n)n)n; SecondConjunct ⊳ ˙¬ ( ˙¬ ((\underline{(y2)} + (−u(rec((1 + 1) + 1) ∗ \underline{z}))) <= \underline{(y1)} ⇒ ˙¬ ( ˙¬ ((\underline{(y2)} + (−u(rec((1 + 1) + 1) ∗ \underline{z}))) = \underline{(y1)})n)n)n ⇒ ˙¬ ( ˙¬ (\underline{(y1)} <= (\underline{(y2)} + (rec((1 + 1) + 1) ∗ \underline{z})) ⇒ ˙¬ ( ˙¬ (\underline{(y1)} = (\underline{(y2)} + (rec((1 + 1) + 1) ∗ \underline{z})))n)n)n)n)n ≫ ˙¬ (\underline{(y1)} <= (\underline{(y2)} + (rec((1 + 1) + 1) ∗ \underline{z})) ⇒ ˙¬ ( ˙¬ (\underline{(y1)} = (\underline{(y2)} + (rec((1 + 1) + 1) ∗ \underline{z})))n)n)n; PositiveToLeft(Less) ⊳ ˙¬ (\underline{(y1)} <= (\underline{(y2)}+(rec((1+1)+1)∗\underline{z})) ⇒ ˙¬ ( ˙¬ (\underline{(y1)} = (\underline{(y2)}+(rec((1+1)+1)∗\underline{z})))n)n)n ≫ ˙¬ ((\underline{(y1)}+(−u(rec((1+1)+1)∗\underline{z}))) <= \underline{(y2)} ⇒ ˙¬ ( ˙¬ ((\underline{(y1)}+(−u(rec((1+1)+1)∗ \underline{z}))) = \underline{(y2)})n)n)n; LessAddition⊳ ˙¬ ((\underline{(y1)}+(−u(rec((1+1)+1)∗\underline{z}))) <= \underline{(y2)} ⇒ ˙¬ ( ˙¬ ((\underline{(y1)}+(−u(rec((1+1)+1)∗\underline{z}))) = \underline{(y2)})n)n)n ≫ ˙¬ (((\underline{(y1)}+(−u(rec((1+ 1)+1)∗\underline{z})))+(−u(rec((1+1)+1)∗\underline{z}))) <= (\underline{(y2)}+(−u(rec((1+1)+1)∗\underline{z}))) ⇒ ˙¬ ( ˙¬ (((\underline{(y1)} + (−u(rec((1 + 1) + 1) ∗ \underline{z}))) + (−u(rec((1 + 1) + 1) ∗ \underline{z}))) = (\underline{(y2)} + (−u(rec((1 + 1) + 1) ∗ \underline{z}))))n)n)n; plusAssociativity ≫ ((\underline{(y1)} + (−u(rec((1 + 1) + 1) ∗ \underline{z}))) + (−u(rec((1 + 1) + 1) ∗ \underline{z}))) = (\underline{(y1)}+((−u(rec((1+1)+1)∗\underline{z}))+(−u(rec((1+1)+1)∗\underline{z})))); −(1/3)x−(1/3)x = −(2/3)x ≫ ((−u(rec((1 + 1) + 1) ∗ \underline{z})) + (−u(rec((1 + 1) + 1) ∗ \underline{z}))) = (−u(((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{z})); EqAdditionLeft ⊳ ((−u(rec((1 + 1) + 1) ∗ \underline{z})) + (−u(rec((1 + 1) + 1) ∗ \underline{z}))) = (−u(((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{z})) ≫ (\underline{(y1)} + ((−u(rec((1 + 1) + 1) ∗ \underline{z})) + (−u(rec((1 + 1) + 1) ∗ \underline{z})))) = (\underline{(y1)} + (−u(((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{z}))); eqTransitivity ⊳ ((\underline{(y1)} + (−u(rec((1 + 1) + 1) ∗ \underline{z}))) + (−u(rec((1 + 1) + 1) ∗ \underline{z}))) = (\underline{(y1)} + ((−u(rec((1 + 1) + 1) ∗ \underline{z})) + (−u(rec((1 + 1) + 1) ∗ \underline{z})))) ⊳ (\underline{(y1)} + ((−u(rec((1 + 1) + 1) ∗ \underline{z})) + (−u(rec((1 + 1) + 1) ∗ \underline{z})))) = (\underline{(y1)} + (−u(((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{z}))) ≫ ((\underline{(y1)} + (−u(rec((1 + 1) + 1) ∗ \underline{z}))) + (−u(rec((1 + 1) + 1) ∗ \underline{z}))) = (\underline{(y1)} + (−u(((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{z}))); SubLessLeft ⊳ ((\underline{(y1)} + (−u(rec((1 + 1) + 1) ∗ \underline{z}))) + (−u(rec((1 + 1) + 1) ∗ \underline{z}))) = (\underline{(y1)} + (−u(((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{z}))) ⊳ ˙¬ (((\underline{(y1)} + (−u(rec((1 + 1) + 1) ∗ \underline{z}))) + (−u(rec((1 + 1) + 1) ∗ \underline{z}))) <= (\underline{(y2)} + (−u(rec((1 + 1) + 1) ∗ \underline{z}))) ⇒ ˙¬ ( ˙¬ (((\underline{(y1)} + (−u(rec((1 + 1) + 1) ∗ \underline{z}))) + (−u(rec((1 + 1) + 1) ∗ \underline{z}))) = (\underline{(y2)} + (−u(rec((1 + 1) + 1) ∗ \underline{z}))))n)n)n ≫ ˙¬ ((\underline{(y1)} + (−u(((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{z}))) <= (\underline{(y2)} + (−u(rec((1 + 1) + 1) ∗ \underline{z}))) ⇒ ˙¬ ( ˙¬ ((\underline{(y1)} + (−u(((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{z}))) = (\underline{(y2)} + (−u(rec((1 + 1) + 1) ∗ \underline{z}))))n)n)n; LessLeqTransitivity ⊳ ˙¬ (\underline{(x2)} <= (\underline{(x1)}+(rec((1+1)+1)∗\underline{z})) ⇒ ˙¬ ( ˙¬ (\underline{(x2)} = (\underline{(x1)}+(rec((1+1)+1)∗\underline{z})))n)n)n⊳ (\underline{(x1)} + (rec((1 + 1) + 1) ∗ \underline{z})) <= (\underline{(y1)} + (−u(((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{z}))) ≫ ˙¬ (\underline{(x2)} <= (\underline{(y1)} + (−u(((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{z}))) ⇒ ˙¬ ( ˙¬ (\underline{(x2)} = (\underline{(y1)}+(−u(((1+1)∗rec((1+1)+1))∗\underline{z}))))n)n)n; LessTransitivity ⊳ ˙¬ (\underline{(x2)} <= (\underline{(y1)} + (−u(((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{z}))) ⇒ ˙¬ ( ˙¬ (\underline{(x2)} = (\underline{(y1)} + (−u(((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{z}))))n)n)n ⊳ ˙¬ ((\underline{(y1)} + (−u(((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{z}))) <= (\underline{(y2)} + (−u(rec((1 + 1) + 1) ∗ \underline{z}))) ⇒ ˙¬ ( ˙¬ ((\underline{(y1)} + (−u(((1 + 1) ∗ rec((1 + 1) + 1)) ∗ \underline{z}))) = (\underline{(y2)} + (−u(rec((1 + 1) + 1) ∗ \underline{z}))))n)n)n ≫ ˙¬ (\underline{(x2)} <= (\underline{(y2)} + (−u(rec((1 + 1) + 1) ∗ \underline{z}))) ⇒ ˙¬ ( ˙¬ (\underline{(x2)} = (\underline{(y2)} + (−u(rec((1 + 1) + 1) ∗ \underline{z}))))n)n)n; LessLeq ⊳ ˙¬ (\underline{(x2)} <= (\underline{(y2)} + (−u(rec((1 + 1) + 1) ∗ \underline{z}))) ⇒ ˙¬ ( ˙¬ (\underline{(x2)} = (\underline{(y2)} + (−u(rec((1 + 1) + 1) ∗ \underline{z}))))n)n)n ≫ \underline{(x2)} <= (\underline{(y2)} + (−u(rec((1 + 1) + 1) ∗ \underline{z})))⌉, p_{0}, c)]

[PreserveLessGreater →^{stmt} SystemQ ⊢ ∀\underline{(x1)}: ∀\underline{(x2)}: ∀\underline{(y1)}: ∀\underline{(y2)}: ∀\underline{z}: \underline{(x1)} <= (\underline{(y1)} + (−u\underline{z})) ⊢ ˙¬ (|(\underline{(x1)} + (−u\underline{(x2)}))| <= (rec((1 + 1) + 1) ∗ \underline{z}) ⇒ ˙¬ ( ˙¬ (|(\underline{(x1)} + (−u\underline{(x2)}))| = (rec((1 + 1) + 1) ∗ \underline{z}))n)n)n ⊢ ˙¬ (|(\underline{(y1)} + (−u\underline{(y2)}))| <= (rec((1 + 1) + 1) ∗ \underline{z}) ⇒ ˙¬ ( ˙¬ (|(\underline{(y1)} + (−u\underline{(y2)}))| = (rec((1 + 1) + 1) ∗ \underline{z}))n)n)n ⊢ \underline{(x2)} <= (\underline{(y2)} + (−u(rec((1 + 1) + 1) ∗ \underline{z})))]

[PreserveLessGreater →^{tex} “PreserveLessGreater”]

[PreserveLessGreater →^{pyk} “lemma preserveLessGreater”]

ClosetolessIsLess

[ClosetolessIsLess →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(x1)}: ∀\underline{(x2)}: ∀\underline{y}: ∀\underline{z}: \underline{(x1)} <= (\underline{y} + (−u\underline{z})) ⊢ ˙¬ (|(\underline{(x1)} + (−u\underline{(x2)}))| <= (rec((1 + 1) + 1) ∗ \underline{z}) ⇒ ˙¬ ( ˙¬ (|(\underline{(x1)} + (−u\underline{(x2)}))| = (rec((1 + 1) + 1) ∗ \underline{z}))n)n)n ⊢ 0 <= |x| ≫ 0 <= |(\underline{(x1)} + (−u\underline{(x2)}))|; leqLessTransitivity ⊳ 0 <= |(\underline{(x1)} + (−u\underline{(x2)}))| ⊳ ˙¬ (|(\underline{(x1)} + (−u\underline{(x2)}))| <= (rec((1 + 1) + 1) ∗ \underline{z}) ⇒ ˙¬ ( ˙¬ (|(\underline{(x1)} + (−u\underline{(x2)}))| = (rec((1 + 1) + 1) ∗ \underline{z}))n)n)n ≫ ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{z}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{z}))n)n)n; |x − x| = 0 ≫ |(\underline{y} + (−u\underline{y}))| = 0; eqSymmetry ⊳ |(\underline{y} + (−u\underline{y}))| = 0 ≫ 0 = |(\underline{y} + (−u\underline{y}))|; SubLessLeft ⊳ 0 = |(\underline{y} + (−u\underline{y}))| ⊳ ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{z}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1+1)+1)∗\underline{z}))n)n)n ≫ ˙¬ (|(\underline{y}+(−u\underline{y}))| <= (rec((1+1)+1)∗\underline{z}) ⇒ ˙¬ ( ˙¬ (|(\underline{y} + (−u\underline{y}))| = (rec((1 + 1) + 1) ∗ \underline{z}))n)n)n; PreserveLessGreater ⊳ \underline{(x1)} <= (\underline{y} + (−u\underline{z})) ⊳ ˙¬ (|(\underline{(x1)} + (−u\underline{(x2)}))| <= (rec((1 + 1) + 1) ∗ \underline{z}) ⇒ ˙¬ ( ˙¬ (|(\underline{(x1)} + (−u\underline{(x2)}))| = (rec((1 + 1) + 1) ∗ \underline{z}))n)n)n ⊳ ˙¬ (|(\underline{y} + (−u\underline{y}))| <= (rec((1 + 1) + 1) ∗ \underline{z}) ⇒ ˙¬ ( ˙¬ (|(\underline{y} + (−u\underline{y}))| = (rec((1 + 1) + 1) ∗ \underline{z}))n)n)n ≫ \underline{(x2)} <= (\underline{y} + (−u(rec((1 + 1) + 1) ∗ \underline{z})))⌉, p_{0}, c)]

[ClosetolessIsLess →^{stmt} SystemQ ⊢ ∀\underline{(x1)}: ∀\underline{(x2)}: ∀\underline{y}: ∀\underline{z}: \underline{(x1)} <= (\underline{y} + (−u\underline{z})) ⊢ ˙¬ (|(\underline{(x1)} + (−u\underline{(x2)}))| <= (rec((1 + 1) + 1) ∗ \underline{z}) ⇒ ˙¬ ( ˙¬ (|(\underline{(x1)} + (−u\underline{(x2)}))| = (rec((1 + 1) + 1) ∗ \underline{z}))n)n)n ⊢ \underline{(x2)} <= (\underline{y} + (−u(rec((1 + 1) + 1) ∗ \underline{z})))]

[ClosetolessIsLess →^{tex} “ClosetolessIsLess”]

[ClosetolessIsLess →^{pyk} “lemma closetolessIsLess”]

SubLessLeft(F)

[SubLessLeft(F) →^{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}] + (−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{(fz)}[\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}] + (−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; 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}] + (−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{(n1)}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{(n1)} <= \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; A4 @(rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⊳ ∀_{obj}\overline{(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{(n1)}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ⇒ \overline{(n1)} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = (rec((1 + 1) + 1) ∗ \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{(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; 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{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(n2)}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{(n2)} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n; ∀\underline{(fx)}: ∀\underline{(fx)}: ∀\underline{(fz)}: ∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{(n2)} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n ⊢ A4 @ \overline{m} ⊳ ∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{(n2)} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n ≫ ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{(n2)} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n; FirstConjunct ⊳ ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{(n2)} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n ≫ ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n; PositiveDividedBy3 ⊳ ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ≫ ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n; ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀_{obj}\overline{m}: ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ⇒ \overline{(n1)} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ⊢ ∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{(n2)} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n ⊢ if(\overline{(n2)} <= \overline{(n1)}, \overline{(n1)}, \overline{(n2)}) <= \overline{m} ⊢ MaxLeq(1) ≫ \overline{(n1)} <= if(\overline{(n2)} <= \overline{(n1)}, \overline{(n1)}, \overline{(n2)}); leqTransitivity ⊳ \overline{(n1)} <= if(\overline{(n2)} <= \overline{(n1)}, \overline{(n1)}, \overline{(n2)}) ⊳ if(\overline{(n2)} <= \overline{(n1)}, \overline{(n1)}, \overline{(n2)}) <= \overline{m} ≫ \overline{(n1)} <= \overline{m}; MaxLeq(2) ≫ \overline{(n2)} <= if(\overline{(n2)} <= \overline{(n1)}, \overline{(n1)}, \overline{(n2)}); leqTransitivity ⊳ \overline{(n2)} <= if(\overline{(n2)} <= \overline{(n1)}, \overline{(n1)}, \overline{(n2)}) ⊳ if(\overline{(n2)} <= \overline{(n1)}, \overline{(n1)}, \overline{(n2)}) <= \overline{m} ≫ \overline{(n2)} <= \overline{m}; A4 @ \overline{m} ⊳ ∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{(n2)} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n ≫ ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{(n2)} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n; FirstConjunct ⊳ ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{(n2)} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n ≫ ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n; SecondConjunct ⊳ ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{(n2)} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n ≫ \overline{(n2)} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})); MP ⊳ \overline{(n2)} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})) ⊳ \overline{(n2)} <= \overline{m} ≫ \underline{(fx)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})); A4 @ \overline{m} ⊳ ∀_{obj}\overline{m}: ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ⇒ \overline{(n1)} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ≫ ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ⇒ \overline{(n1)} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n; PositiveDividedBy3 ⊳ ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ≫ ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n; MP2 ⊳ ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ⇒ \overline{(n1)} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ⊳ ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ⊳ \overline{(n1)} <= \overline{m} ≫ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n; ClosetolessIsLess ⊳ \underline{(fx)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})) ⊳ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ≫ \underline{(fy)}[\overline{m}] <= (\underline{(fz)}[\overline{m}]+(−u(rec((1+1)+1)∗\overline{(ϵ)}))); Ded⊳∀\underline{(fx)}: ∀\underline{(fx)}: ∀\underline{(fz)}: ∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{(n2)} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n ⊢ ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ≫ ∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{(n2)} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n ⇒ ˙¬ (0 <= (rec((1+1)+1)∗\overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1+1)+1)∗\overline{(ϵ)}))n)n)n; TwiceExistMP⊳ ∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{(n2)} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n ⇒ ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(n2)}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{(n2)} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n; Ded ⊳ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀_{obj}\overline{m}: ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ⇒ \overline{(n1)} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = (rec((1+1)+1)∗\overline{(ϵ)}))n)n)n ⊢ ∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{(n2)} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n ⊢ if(\overline{(n2)} <= \overline{(n1)}, \overline{(n1)}, \overline{(n2)}) <= \overline{m} ⊢ \underline{(fy)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u(rec((1 + 1) + 1) ∗ \overline{(ϵ)}))) ≫ ∀_{obj}\overline{m}: ˙¬ (0 <= (rec((1+1)+1)∗\overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1+1)+1)∗\overline{(ϵ)}))n)n)n ⇒ \overline{(n1)} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ⇒ ∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{(n2)} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n ⇒ if(\overline{(n2)} <= \overline{(n1)}, \overline{(n1)}, \overline{(n2)}) <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u(rec((1 + 1) + 1) ∗ \overline{(ϵ)}))); ExistMP ⊳ ∀_{obj}\overline{m}: ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ⇒ \overline{(n1)} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ⇒ ∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{(n2)} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n ⇒ if(\overline{(n2)} <= \overline{(n1)}, \overline{(n1)}, \overline{(n2)}) <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u(rec((1 + 1) + 1) ∗ \overline{(ϵ)}))) ⊳ ˙¬ (∀_{obj}\overline{(n1)}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ⇒ \overline{(n1)} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n)n)n ≫ ∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{(n2)} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n ⇒ if(\overline{(n2)} <= \overline{(n1)}, \overline{(n1)}, \overline{(n2)}) <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u(rec((1 + 1) + 1) ∗ \overline{(ϵ)}))); TwiceExistMP ⊳ ∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{(n2)} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n ⇒ if(\overline{(n2)} <= \overline{(n1)}, \overline{(n1)}, \overline{(n2)}) <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u(rec((1 + 1) + 1) ∗ \overline{(ϵ)}))) ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(n2)}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{(n2)} <= \overline{m} ⇒ \underline{(fx)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ if(\overline{(n2)} <= \overline{(n1)}, \overline{(n1)}, \overline{(n2)}) <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u(rec((1 + 1) + 1) ∗ \overline{(ϵ)}))); JoinConjuncts ⊳ ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ⊳ if(\overline{(n2)} <= \overline{(n1)}, \overline{(n1)}, \overline{(n2)}) <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u(rec((1 + 1) + 1) ∗ \overline{(ϵ)}))) ≫ ˙¬ ( ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ⇒ ˙¬ (if(\overline{(n2)} <= \overline{(n1)}, \overline{(n1)}, \overline{(n2)}) <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u(rec((1 + 1) + 1) ∗ \overline{(ϵ)}))))n)n; Gen ⊳ ˙¬ ( ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ⇒ ˙¬ (if(\overline{(n2)} <= \overline{(n1)}, \overline{(n1)}, \overline{(n2)}) <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fz)}[\overline{m}]+(−u(rec((1+1)+1)∗\overline{(ϵ)}))))n)n ≫ ∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ⇒ ˙¬ (if(\overline{(n2)} <= \overline{(n1)}, \overline{(n1)}, \overline{(n2)}) <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u(rec((1 + 1) + 1) ∗ \overline{(ϵ)}))))n)n; IntroExist @ if(\overline{(n2)} <= \overline{(n1)}, \overline{(n1)}, \overline{(n2)})⊳∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= (rec((1+1)+1)∗\overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1+ 1) + 1) ∗ \overline{(ϵ)}))n)n)n ⇒ ˙¬ (if(\overline{(n2)} <= \overline{(n1)}, \overline{(n1)}, \overline{(n2)}) <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u(rec((1 + 1) + 1) ∗ \overline{(ϵ)}))))n)n ≫ ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u(rec((1 + 1) + 1) ∗ \overline{(ϵ)}))))n)n)n)n; IntroExist @(rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⊳ ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fy)}[\overline{m}] <= (\underline{(fz)}[\overline{m}] + (−u(rec((1 + 1) + 1) ∗ \overline{(ϵ)}))))n)n)n)n ≫ ˙¬ (∀_{obj}\overline{(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; Repetition ⊳ ˙¬ (∀_{obj}\overline{(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⌉, p_{0}, c)]

[SubLessLeft(F) →^{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}] + (−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{(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]

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

[SubLessLeft(F) →^{pyk} “lemma subLessLeft(F)”]

SubLessLeft(R)

[SubLessLeft(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 ⊢ 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; 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{(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; SubLessLeft(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}] + (−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{(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; Repetition ⊳ ˙¬ (∀_{obj}\overline{(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⌉, p_{0}, c)]

[SubLessLeft(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 ⊢ ˙¬ (∀_{obj}\overline{(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]

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

[SubLessLeft(R) →^{pyk} “lemma subLessLeft(R)”]

ClosetogreaterIsGreater

[ClosetogreaterIsGreater →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{(y1)}: ∀\underline{(y2)}: ∀\underline{z}: \underline{x} <= (\underline{(y1)} + (−u\underline{z})) ⊢ ˙¬ (|(\underline{(y1)} + (−u\underline{(y2)}))| <= (rec((1 + 1) + 1) ∗ \underline{z}) ⇒ ˙¬ ( ˙¬ (|(\underline{(y1)} + (−u\underline{(y2)}))| = (rec((1 + 1) + 1) ∗ \underline{z}))n)n)n ⊢ 0 <= |x| ≫ 0 <= |(\underline{(y1)} + (−u\underline{(y2)}))|; leqLessTransitivity ⊳ 0 <= |(\underline{(y1)} + (−u\underline{(y2)}))| ⊳ ˙¬ (|(\underline{(y1)} + (−u\underline{(y2)}))| <= (rec((1 + 1) + 1) ∗ \underline{z}) ⇒ ˙¬ ( ˙¬ (|(\underline{(y1)} + (−u\underline{(y2)}))| = (rec((1 + 1) + 1) ∗ \underline{z}))n)n)n ≫ ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{z}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \underline{z}))n)n)n; |x − x| = 0 ≫ |(\underline{x} + (−u\underline{x}))| = 0; eqSymmetry ⊳ |(\underline{x} + (−u\underline{x}))| = 0 ≫ 0 = |(\underline{x} + (−u\underline{x}))|; SubLessLeft ⊳ 0 = |(\underline{x} + (−u\underline{x}))| ⊳ ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \underline{z}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1+1)+1)∗\underline{z}))n)n)n ≫ ˙¬ (|(\underline{x}+(−u\underline{x}))| <= (rec((1+1)+1)∗\underline{z}) ⇒ ˙¬ ( ˙¬ (|(\underline{x} + (−u\underline{x}))| = (rec((1 + 1) + 1) ∗ \underline{z}))n)n)n; PreserveLessGreater ⊳ \underline{x} <= (\underline{(y1)} + (−u\underline{z})) ⊳ ˙¬ (|(\underline{x} + (−u\underline{x}))| <= (rec((1 + 1) + 1) ∗ \underline{z}) ⇒ ˙¬ ( ˙¬ (|(\underline{x} + (−u\underline{x}))| = (rec((1 + 1) + 1) ∗ \underline{z}))n)n)n ⊳ ˙¬ (|(\underline{(y1)} + (−u\underline{(y2)}))| <= (rec((1 + 1) + 1) ∗ \underline{z}) ⇒ ˙¬ ( ˙¬ (|(\underline{(y1)} + (−u\underline{(y2)}))| = (rec((1 + 1) + 1) ∗ \underline{z}))n)n)n ≫ \underline{x} <= (\underline{(y2)} + (−u(rec((1 + 1) + 1) ∗ \underline{z})))⌉, p_{0}, c)]

[ClosetogreaterIsGreater →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{(y1)}: ∀\underline{(y2)}: ∀\underline{z}: \underline{x} <= (\underline{(y1)}+(−u\underline{z})) ⊢ ˙¬ (|(\underline{(y1)}+(−u\underline{(y2)}))| <= (rec((1+1)+1)∗\underline{z}) ⇒ ˙¬ ( ˙¬ (|(\underline{(y1)}+ (−u\underline{(y2)}))| = (rec((1+1)+1)∗\underline{z}))n)n)n ⊢ \underline{x} <= (\underline{(y2)}+(−u(rec((1+1)+1)∗\underline{z})))]

[ClosetogreaterIsGreater →^{tex} “ClosetogreaterIsGreater”] [ClosetogreaterIsGreater →^{pyk} “lemma closetogreaterIsGreater”]

SubLessRight(F)

[SubLessRight(F) →^{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}] + (−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{(fz)}[\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}] + (−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; 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}] + (−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{(n1)}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ \overline{(n1)} <= \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; A4 @(rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⊳ ∀_{obj}\overline{(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{(n1)}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ⇒ \overline{(n1)} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = (rec((1 + 1) + 1) ∗ \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{(fz)}[\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{(fz)}[\overline{m}] <= (\underline{(fx)}[\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{(fz)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(n2)}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{(n2)} <= \overline{m} ⇒ \underline{(fz)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n; ∀\underline{(fx)}: ∀\underline{(fx)}: ∀\underline{(fz)}: ∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{(n2)} <= \overline{m} ⇒ \underline{(fz)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n ⊢ A4 @ \overline{m} ⊳ ∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{(n2)} <= \overline{m} ⇒ \underline{(fz)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n ≫ ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{(n2)} <= \overline{m} ⇒ \underline{(fz)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n; FirstConjunct ⊳ ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{(n2)} <= \overline{m} ⇒ \underline{(fz)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n ≫ ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n; PositiveDividedBy3 ⊳ ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ≫ ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n; ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀_{obj}\overline{m}: ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ⇒ \overline{(n1)} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ⊢ ∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{(n2)} <= \overline{m} ⇒ \underline{(fz)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n ⊢ if(\overline{(n2)} <= \overline{(n1)}, \overline{(n1)}, \overline{(n2)}) <= \overline{m} ⊢ MaxLeq(1) ≫ \overline{(n1)} <= if(\overline{(n2)} <= \overline{(n1)}, \overline{(n1)}, \overline{(n2)}); leqTransitivity ⊳ \overline{(n1)} <= if(\overline{(n2)} <= \overline{(n1)}, \overline{(n1)}, \overline{(n2)}) ⊳ if(\overline{(n2)} <= \overline{(n1)}, \overline{(n1)}, \overline{(n2)}) <= \overline{m} ≫ \overline{(n1)} <= \overline{m}; MaxLeq(2) ≫ \overline{(n2)} <= if(\overline{(n2)} <= \overline{(n1)}, \overline{(n1)}, \overline{(n2)}); leqTransitivity ⊳ \overline{(n2)} <= if(\overline{(n2)} <= \overline{(n1)}, \overline{(n1)}, \overline{(n2)}) ⊳ if(\overline{(n2)} <= \overline{(n1)}, \overline{(n1)}, \overline{(n2)}) <= \overline{m} ≫ \overline{(n2)} <= \overline{m}; A4 @ \overline{m} ⊳ ∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{(n2)} <= \overline{m} ⇒ \underline{(fz)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n ≫ ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{(n2)} <= \overline{m} ⇒ \underline{(fz)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n; FirstConjunct ⊳ ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{(n2)} <= \overline{m} ⇒ \underline{(fz)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n ≫ ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n; SecondConjunct ⊳ ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{(n2)} <= \overline{m} ⇒ \underline{(fz)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n ≫ \overline{(n2)} <= \overline{m} ⇒ \underline{(fz)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})); MP ⊳ \overline{(n2)} <= \overline{m} ⇒ \underline{(fz)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})) ⊳ \overline{(n2)} <= \overline{m} ≫ \underline{(fz)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})); A4 @ \overline{m} ⊳ ∀_{obj}\overline{m}: ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ⇒ \overline{(n1)} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ≫ ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ⇒ \overline{(n1)} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n; PositiveDividedBy3 ⊳ ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ≫ ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n; MP2 ⊳ ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ⇒ \overline{(n1)} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ⊳ ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ⊳ \overline{(n1)} <= \overline{m} ≫ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n; ClosetogreaterIsGreater ⊳ \underline{(fz)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})) ⊳ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ≫ \underline{(fz)}[\overline{m}] <= (\underline{(fy)}[\overline{m}]+(−u(rec((1+1)+1)∗\overline{(ϵ)}))); Ded⊳∀\underline{(fx)}: ∀\underline{(fx)}: ∀\underline{(fz)}: ∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{(n2)} <= \overline{m} ⇒ \underline{(fz)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n ⊢ ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ≫ ∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{(n2)} <= \overline{m} ⇒ \underline{(fz)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n ⇒ ˙¬ (0 <= (rec((1+1)+1)∗\overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1+1)+1)∗\overline{(ϵ)}))n)n)n; TwiceExistMP⊳ ∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{(n2)} <= \overline{m} ⇒ \underline{(fz)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n ⇒ ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(n2)}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{(n2)} <= \overline{m} ⇒ \underline{(fz)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n; Ded ⊳ ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(fz)}: ∀_{obj}\overline{m}: ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ⇒ \overline{(n1)} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = (rec((1+1)+1)∗\overline{(ϵ)}))n)n)n ⊢ ∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{(n2)} <= \overline{m} ⇒ \underline{(fz)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n ⊢ if(\overline{(n2)} <= \overline{(n1)}, \overline{(n1)}, \overline{(n2)}) <= \overline{m} ⊢ \underline{(fz)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u(rec((1 + 1) + 1) ∗ \overline{(ϵ)}))) ≫ ∀_{obj}\overline{m}: ˙¬ (0 <= (rec((1+1)+1)∗\overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1+1)+1)∗\overline{(ϵ)}))n)n)n ⇒ \overline{(n1)} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ⇒ ∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{(n2)} <= \overline{m} ⇒ \underline{(fz)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n ⇒ if(\overline{(n2)} <= \overline{(n1)}, \overline{(n1)}, \overline{(n2)}) <= \overline{m} ⇒ \underline{(fz)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u(rec((1 + 1) + 1) ∗ \overline{(ϵ)}))); ExistMP ⊳ ∀_{obj}\overline{m}: ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ⇒ \overline{(n1)} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ⇒ ∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{(n2)} <= \overline{m} ⇒ \underline{(fz)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n ⇒ if(\overline{(n2)} <= \overline{(n1)}, \overline{(n1)}, \overline{(n2)}) <= \overline{m} ⇒ \underline{(fz)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u(rec((1 + 1) + 1) ∗ \overline{(ϵ)}))) ⊳ ˙¬ (∀_{obj}\overline{(n1)}: ˙¬ (∀_{obj}\overline{m}: ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ⇒ \overline{(n1)} <= \overline{m} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n)n)n ≫ ∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{(n2)} <= \overline{m} ⇒ \underline{(fz)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n ⇒ if(\overline{(n2)} <= \overline{(n1)}, \overline{(n1)}, \overline{(n2)}) <= \overline{m} ⇒ \underline{(fz)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u(rec((1 + 1) + 1) ∗ \overline{(ϵ)}))); TwiceExistMP ⊳ ∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{(n2)} <= \overline{m} ⇒ \underline{(fz)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n ⇒ if(\overline{(n2)} <= \overline{(n1)}, \overline{(n1)}, \overline{(n2)}) <= \overline{m} ⇒ \underline{(fz)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u(rec((1 + 1) + 1) ∗ \overline{(ϵ)}))) ⊳ ˙¬ (∀_{obj}\overline{(ϵ)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(n2)}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ ˙¬ (\overline{(n2)} <= \overline{m} ⇒ \underline{(fz)}[\overline{m}] <= (\underline{(fx)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n ≫ if(\overline{(n2)} <= \overline{(n1)}, \overline{(n1)}, \overline{(n2)}) <= \overline{m} ⇒ \underline{(fz)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u(rec((1 + 1) + 1) ∗ \overline{(ϵ)}))); JoinConjuncts ⊳ ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ⊳ if(\overline{(n2)} <= \overline{(n1)}, \overline{(n1)}, \overline{(n2)}) <= \overline{m} ⇒ \underline{(fz)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u(rec((1 + 1) + 1) ∗ \overline{(ϵ)}))) ≫ ˙¬ ( ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ⇒ ˙¬ (if(\overline{(n2)} <= \overline{(n1)}, \overline{(n1)}, \overline{(n2)}) <= \overline{m} ⇒ \underline{(fz)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u(rec((1 + 1) + 1) ∗ \overline{(ϵ)}))))n)n; Gen ⊳ ˙¬ ( ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ⇒ ˙¬ (if(\overline{(n2)} <= \overline{(n1)}, \overline{(n1)}, \overline{(n2)}) <= \overline{m} ⇒ \underline{(fz)}[\overline{m}] <= (\underline{(fy)}[\overline{m}]+(−u(rec((1+1)+1)∗\overline{(ϵ)}))))n)n ≫ ∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ⇒ ˙¬ (if(\overline{(n2)} <= \overline{(n1)}, \overline{(n1)}, \overline{(n2)}) <= \overline{m} ⇒ \underline{(fz)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u(rec((1 + 1) + 1) ∗ \overline{(ϵ)}))))n)n; IntroExist @ if(\overline{(n2)} <= \overline{(n1)}, \overline{(n1)}, \overline{(n2)})⊳∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= (rec((1+1)+1)∗\overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1+ 1) + 1) ∗ \overline{(ϵ)}))n)n)n ⇒ ˙¬ (if(\overline{(n2)} <= \overline{(n1)}, \overline{(n1)}, \overline{(n2)}) <= \overline{m} ⇒ \underline{(fz)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u(rec((1 + 1) + 1) ∗ \overline{(ϵ)}))))n)n ≫ ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fz)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u(rec((1 + 1) + 1) ∗ \overline{(ϵ)}))))n)n)n)n; IntroExist @(rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⊳ ˙¬ (∀_{obj}\overline{n}: ˙¬ (∀_{obj}\overline{m}: ˙¬ ( ˙¬ (0 <= (rec((1 + 1) + 1) ∗ \overline{(ϵ)}) ⇒ ˙¬ ( ˙¬ (0 = (rec((1 + 1) + 1) ∗ \overline{(ϵ)}))n)n)n ⇒ ˙¬ (\overline{n} <= \overline{m} ⇒ \underline{(fz)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u(rec((1 + 1) + 1) ∗ \overline{(ϵ)}))))n)n)n)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; Repetition ⊳ ˙¬ (∀_{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 ≫ ˙¬ (∀_{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⌉, p_{0}, c)]

[SubLessRight(F) →^{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}] + (−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{(fz)}[\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{(fz)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n]

[SubLessRight(F) →^{tex} “SubLessRight(F)”] [SubLessRight(F) →^{pyk} “lemma subLessRight(F)”]

SubLessRight(R)

[SubLessRight(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 ⊢ 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; Repetition ⊳ ˙¬ (∀_{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 ≫ ˙¬ (∀_{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; SubLessRight(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}] + (−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{(fz)}[\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{(fz)}[\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{(fz)}[\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{(fz)}[\overline{m}] <= (\underline{(fy)}[\overline{m}] + (−u\overline{(ϵ)})))n)n)n)n)n)n⌉, p_{0}, c)]

[SubLessRight(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 ⊢ ˙¬ (∀_{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]

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

[SubLessRight(R) →^{pyk} “lemma subLessRight(R)”]

plus0Left

[plus0Left →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: plus0 ≫ (\underline{x} + 0) = \underline{x}; plusCommutativity ≫ (0 + \underline{x}) = (\underline{x} + 0); eqTransitivity ⊳ (0 + \underline{x}) = (\underline{x} + 0) ⊳ (\underline{x} + 0) = \underline{x} ≫ (0 + \underline{x}) = \underline{x}⌉, p_{0}, c)]

[plus0Left →^{stmt} SystemQ ⊢ ∀\underline{x}: (0 + \underline{x}) = \underline{x}]

[plus0Left →^{tex} “plus0Left”]

[plus0Left →^{pyk} “lemma plus0Left”]

times1Left

[times1Left →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: times1 ≫ (\underline{x} ∗ 1) = \underline{x}; timesCommutativity ≫ (1 ∗ \underline{x}) = (\underline{x} ∗ 1); eqTransitivity ⊳ (1 ∗ \underline{x}) = (\underline{x} ∗ 1) ⊳ (\underline{x} ∗ 1) = \underline{x} ≫ (1 ∗ \underline{x}) = \underline{x}⌉, p_{0}, c)]

[times1Left →^{stmt} SystemQ ⊢ ∀\underline{x}: (1 ∗ \underline{x}) = \underline{x}]

[times1Left →^{tex} “times1Left”]

[times1Left →^{pyk} “lemma times1Left”]

EqAdditionLeft

[EqAdditionLeft →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ eqAddition ⊳ \underline{x} = \underline{y} ≫ (\underline{x} + \underline{z}) = (\underline{y} + \underline{z}); plusCommutativity ≫ (\underline{z} + \underline{x}) = (\underline{x} + \underline{z}); plusCommutativity ≫ (\underline{y} + \underline{z}) = (\underline{z} + \underline{y}); eqTransitivity4 ⊳ (\underline{z} + \underline{x}) = (\underline{x} + \underline{z}) ⊳ (\underline{x} + \underline{z}) = (\underline{y} + \underline{z}) ⊳ (\underline{y} + \underline{z}) = (\underline{z} + \underline{y}) ≫ (\underline{z} + \underline{x}) = (\underline{z} + \underline{y})⌉, p_{0}, c)]

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

[EqAdditionLeft →^{tex} “EqAdditionLeft”]

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

EqMultiplicationLeft

[EqMultiplicationLeft →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ eqMultiplication ⊳ \underline{x} = \underline{y} ≫ (\underline{x} ∗ \underline{z}) = (\underline{y} ∗ \underline{z}); timesCommutativity ≫ (\underline{z} ∗ \underline{x}) = (\underline{x} ∗ \underline{z}); timesCommutativity ≫ (\underline{y} ∗ \underline{z}) = (\underline{z} ∗ \underline{y}); eqTransitivity4 ⊳ (\underline{z} ∗ \underline{x}) = (\underline{x} ∗ \underline{z}) ⊳ (\underline{x} ∗ \underline{z}) = (\underline{y} ∗ \underline{z}) ⊳ (\underline{y} ∗ \underline{z}) = (\underline{z} ∗ \underline{y}) ≫ (\underline{z} ∗ \underline{x}) = (\underline{z} ∗ \underline{y})⌉, p_{0}, c)]

[EqMultiplicationLeft →^{stmt} SystemQ ⊢ ∀\underline{x}: ∀\underline{y}: ∀\underline{z}: \underline{x} = \underline{y} ⊢ (\underline{z} ∗ \underline{x}) = (\underline{z} ∗ \underline{y})]

[EqMultiplicationLeft →^{tex} “EqMultiplicationLeft”]

[EqMultiplicationLeft →^{pyk} “lemma eqMultiplicationLeft”]

PlusF(Sym)

[PlusF(Sym) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: PlusF ≫ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(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{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}]); eqSymmetry ⊳ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(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{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}]) ≫ (\underline{(fx)}[\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{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]⌉, p_{0}, c)]

[PlusF(Sym) →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: (\underline{(fx)}[\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{(fx)}[\underline{m}] + \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]]

[PlusF(Sym) →^{tex} “PlusF(Sym)”]

[PlusF(Sym) →^{pyk} “lemma plusF(Sym)”]

TimesF(Sym)

[TimesF(Sym) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: TimesF ≫ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] = (\underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}]); eqSymmetry ⊳ \{ph ∈ \{ph ∈ P(P(Union(\{N, Q\}))) | ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (a_{Ph} = \{\{\overline{(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}] = (\underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}]) ≫ (\underline{(fx)}[\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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]⌉, p_{0}, c)]

[TimesF(Sym) →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: (\underline{(fx)}[\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}: ˙¬ (e_{Ph} = \{\{\underline{m}, \underline{m}\}, \{\underline{m}, (\underline{(fx)}[\underline{m}] ∗ \underline{(fy)}[\underline{m}])\}\})n)n\}[\underline{m}]]

[TimesF(Sym) →^{tex} “TimesF(Sym)”]

[TimesF(Sym) →^{pyk} “lemma timesF(Sym)”]

SameSeries(Gen)

[SameSeries(Gen) →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(sz)}: \underline{m} ∈ N ⊢ \underline{n} ∈ N ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sz)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fy)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sz)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fy)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fy)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fy)})n)n)n)n ⊢ \underline{m} = \underline{n} ⊢ \underline{(fx)} = \underline{(fy)} ⊢ MemberOfSeries ⊳ \underline{m} ∈ N ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sz)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ≫ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fx)}[\underline{m}]\}\} ∈ \underline{(fx)}; SENC1 ⊳ \underline{(fx)} = \underline{(fy)} ⊳ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fx)}[\underline{m}]\}\} ∈ \underline{(fx)} ≫ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fx)}[\underline{m}]\}\} ∈ \underline{(fy)}; MemberOfSeries ⊳ \underline{n} ∈ N ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fy)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sz)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fy)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fy)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fy)})n)n)n)n ≫ \{\{\underline{n}, \underline{n}\}, \{\underline{n}, \underline{(fy)}[\underline{n}]\}\} ∈ \underline{(fy)}; UniqueMember ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fy)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sz)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fy)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fy)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fy)})n)n)n)n ⊳ \{\{\underline{m}, \underline{m}\}, \{\underline{m}, \underline{(fx)}[\underline{m}]\}\} ∈ \underline{(fy)} ⊳ \{\{\underline{n}, \underline{n}\}, \{\underline{n}, \underline{(fy)}[\underline{n}]\}\} ∈ \underline{(fy)} ⊳ \underline{m} = \underline{n} ≫ \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{n}]⌉, p_{0}, c)]

[SameSeries(Gen) →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{n}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀\underline{(sz)}: \underline{m} ∈ N ⊢ \underline{n} ∈ N ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sz)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ⊢ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fy)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ \underline{(sz)})n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fy)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fy)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fy)})n)n)n)n ⊢ \underline{m} = \underline{n} ⊢ \underline{(fx)} = \underline{(fy)} ⊢ \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{n}]]

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

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

EqualsSameF

[EqualsSameF →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)} = \underline{(fy)} ⊢ ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⊢ 0 <= \overline{m} ⊢ NatType ≫ \overline{m} ∈ N; SeriesType ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n; SeriesType ≫ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fy)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fy)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fy)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fy)})n)n)n)n; eqReflexivity ≫ \overline{m} = \overline{m}; SameSeries(Gen) ⊳ \overline{m} ∈ N ⊳ \overline{m} ∈ N ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fx)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fx)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fx)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fx)})n)n)n)n ⊳ ˙¬ ( ˙¬ (∀_{obj}\overline{(r1)}: \overline{(r1)} ∈ \underline{(fy)} ⇒ ˙¬ (∀_{obj}\overline{(op1)}: ˙¬ ( ˙¬ (∀_{obj}\overline{(op2)}: ˙¬ ( ˙¬ ( ˙¬ (\overline{(op1)} ∈ N ⇒ ˙¬ (\overline{(op2)} ∈ Q)n)n ⇒ ˙¬ (\overline{(r1)} = \{\{\overline{(op1)}, \overline{(op1)}\}, \{\overline{(op1)}, \overline{(op2)}\}\})n)n)n)n)n)n ⇒ ˙¬ (∀_{obj}\overline{(f1)}: ∀_{obj}\overline{(f2)}: ∀_{obj}\overline{(f3)}: ∀_{obj}\overline{(f4)}: \{\{\overline{(f1)}, \overline{(f1)}\}, \{\overline{(f1)}, \overline{(f2)}\}\} ∈ \underline{(fy)} ⇒ \{\{\overline{(f3)}, \overline{(f3)}\}, \{\overline{(f3)}, \overline{(f4)}\}\} ∈ \underline{(fy)} ⇒ \overline{(f1)} = \overline{(f3)} ⇒ \overline{(f2)} = \overline{(f4)})n)n ⇒ ˙¬ (∀_{obj}\overline{(s1)}: \overline{(s1)} ∈ N ⇒ ˙¬ (∀_{obj}\overline{(s2)}: ˙¬ (\{\{\overline{(s1)}, \overline{(s1)}\}, \{\overline{(s1)}, \overline{(s2)}\}\} ∈ \underline{(fy)})n)n)n)n ⊳ \overline{m} = \overline{m} ⊳ \underline{(fx)} = \underline{(fy)} ≫ \underline{(fx)}[\overline{m}] = \underline{(fy)}[\overline{m}]; PositiveToLeft(Eq)(1term) ⊳ \underline{(fx)}[\overline{m}] = \underline{(fy)}[\overline{m}] ≫ (\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}])) = 0; SameNumerical ⊳ (\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}])) = 0 ≫ |(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = |0|; |0| = 0 ≫ |0| = 0; eqTransitivity ⊳ |(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| = |0| ⊳ |0| = 0 ≫ |(\underline{(fx)}[\overline{m}]+(−u\underline{(fy)}[\overline{m}]))| = 0; eqSymmetry ⊳|(\underline{(fx)}[\overline{m}]+(−u\underline{(fy)}[\overline{m}]))| = 0 ≫ 0 = |(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))|; SubLessLeft ⊳ 0 = |(\underline{(fx)}[\overline{m}] + (−u\underline{(fy)}[\overline{m}]))| ⊳ ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ≫ ˙¬ (|(\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; ∀\underline{(fx)}: ∀\underline{(fy)}: Ded⊳∀\underline{(fx)}: ∀\underline{(fy)}: \underline{(fx)} = \underline{(fy)} ⊢ ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⊢ 0 <= \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 ≫ \underline{(fx)} = \underline{(fy)} ⇒ ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ 0 <= \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; \underline{(fx)} = \underline{(fy)} ⊢ MP ⊳ \underline{(fx)} = \underline{(fy)} ⇒ ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ 0 <= \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 ⊳ \underline{(fx)} = \underline{(fy)} ≫ ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ 0 <= \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; Gen ⊳ ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ 0 <= \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 ≫ ∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ 0 <= \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; IntroExist @ 0 ⊳ ∀_{obj}\overline{m}: ˙¬ (0 <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (0 = \overline{(ϵ)})n)n)n ⇒ 0 <= \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 ≫ ˙¬ (∀_{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; Gen ⊳ ˙¬ (∀_{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; 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 ≫ ∀_{obj}\overline{(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)]

[EqualsSameF →^{stmt} SystemQ ⊢ ∀\underline{(fx)}: ∀\underline{(fy)}: \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] [EqualsSameF →^{tex} “EqualsSameF”]

[EqualsSameF →^{pyk} “lemma equalsSameF”]

LeqReflexivity(R)

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

[LeqReflexivity(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}] <= (\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} ⇒ ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| <= \overline{(ϵ)} ⇒ ˙¬ ( ˙¬ (|(\underline{(fx)}[\overline{m}] + (−ud_{Ph}[\overline{m}]))| = \overline{(ϵ)})n)n)n)n)n\}]

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

[LeqReflexivity(R) →^{pyk} “lemma leqReflexivity(R)”]

Tester1

[Tester1 →^{proof} λc.λx.P(⌈SystemQ ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀_{obj}\underline{m}: \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}] ⊢ To = f ⊳ ∀_{obj}\underline{m}: \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}] ≫ \underline{(fx)} = \underline{(fy)}⌉, p_{0}, c)]

[Tester1 →^{stmt} SystemQ ⊢ ∀\underline{m}: ∀\underline{(fx)}: ∀\underline{(fy)}: ∀_{obj}\underline{m}: \underline{(fx)}[\underline{m}] = \underline{(fy)}[\underline{m}] ⊢ \underline{(fx)} = \underline{(fy)}]

[Tester1 →^{tex} “Tester1”]

[Tester1 →^{pyk} “tester1”]

Tester2

[Tester2 →^{tex} “Tester2”]

[Tester2 →^{pyk} “tester2”]

Tester3

[Tester3 →^{tex} “Tester3”]

[Tester3 →^{pyk} “tester3”]

Tester4

[Tester4 →^{tex} “Tester4”] [Tester4 →^{pyk} “tester4”]

Tester5

[Tester5 →^{tex} “Tester5”] [Tester5 →^{pyk} “tester5”]

Tester6

[Tester6 →^{tex} “Tester6”] [Tester6 →^{pyk} “tester6”] The pyk compiler, version 0.grue.20060417+ by Klaus Grue GRD-2006-12-29.UTC:09:42:35.018035 = MJD-54098.TAI:09:43:08.018035 = LGT-4674102188018035e-6