PAGE sup

BIBLIOGRAPHY

"base" nani
keta kute sena seku kuka  kute tuti teke tati tetu
sise sinu seku kuna tetu  kisu kune nane tise kike
kesi seku kana sasu sane  kika sunu nasa natu,

"check" nani
tusi nese tutu kena kane  sine setu siti kene keni
tite tena sasi sasa site  sutu sane nita nini nasi
kana sane suka sane sasu  kika sunu nasa natu,

"kvanti" nani
kuku seka sesi situ tasi
kiti kisa naki nata sike
kisu nenu suna tisu tase
nuse kenu kusa teta nune
kuta suku kaka saku keka
suse sutu nasa natu

PREASSOCIATIVE ""; RR, KVANTI (linie 23...)
"base" bracket " end bracket
"check"   general macro define " as " end define
"kvanti" cdots


""; REGELLEMMAER
"" lemma to!!==

""; UDSAGNSLOGIK
"" prop lemma to negated double imply
"" prop lemma to negated and(1)

""; PRAEDIKATLOGK
"" pred lemma addNegatedAll
"" pred lemma (A)to(~E~)(Imply)
"" pred lemma (E)to(~A~)(Imply)
"" pred lemma (E~)to(~A)(Imply)
"" pred lemma toNegatedAEA

""; MINUS
"" lemma uniqueNegative
"" lemma doubleMinus
"" lemma minusNegated


""; EQ
"" lemma eqReflexivity
"" lemma eqSymmetry
"" lemma eqTransitivity
"" lemma eqTransitivity4
"" lemma eqTransitivity5
"" lemma eqTransitivity6
"" lemma addEquations
"" lemma subtractEquations
"" lemma subtractEquationsLeft
"" lemma multiplyEquations
"" lemma eqNegated
"" lemma positiveToRight(Eq)
"" lemma positiveToLeft(Eq)
"" lemma positiveToLeft(Eq)(1 term)
"" lemma negativeToLeft(Eq)
"" lemma nonreciprocalToRight(Eq)(1 term)
"" lemma distributionOut(Minus)
"" lemma positiveToRight(Eq)(1 term)
"" lemma sameSeries(NumDiff)
"" lemma plusAssociativity(4 terms)

""; NEQ
"" lemma lessNeq
"" lemma neqSymmetry
"" lemma neqNegated
"" lemma subNeqRight
"" lemma subNeqLeft
"" lemma negativeToRight(Neq)(1 term)
"" lemma neqAddition
"" lemma neqMultiplication
"" lemma nonzeroProduct(2)

""; LEQ
"" lemma switchTerms(x<=y-z)

""; LESS
"" lemma negativeToLeft(Less)(1 term)

""; NATURLIGE TAL 
"" lemma +1IsPositive(N)

""; REGNESTYKKER
"" lemma (1/2)(x+y)-x=(1/2)(y-x)
"" lemma y-(1/2)(x+y)=(1/2)(y-x)

""; EXP
"" lemma expZero exact
"" lemma sameExp base
"" lemma sameExp indu
"" lemma sameExp
"" lemma exp(+1)
"" lemma positiveBase base
"" lemma positiveBase indu
"" lemma positiveBase

""; BASE(1/2)SUM
"" lemma base(1/2)Sum zero exact
"" lemma sameBase(1/2)Sum second base
"" lemma sameBase(1/2)Sum second indu
"" lemma sameBase(1/2)Sum second 
"" lemma base(1/2)Sum(+1)
"" lemma base(1/2)Sum exact bound base
"" lemma base(1/2)Sum exact bound indu
"" lemma base(1/2)Sum exact bound
"" lemma base(1/2)Sum bound

""; TELESCOPE
"" lemma UStelescope zero exact
"" lemma sameTelescope second base
"" lemma sameTelescope second indu
"" lemma sameTelescope second
"" lemma UStelescope(+1)
"" lemma telescopeNumerical base
"" lemma telescopeNumerical indu
"" lemma telescopeNumerical
"" lemma telescopeBound base
"" lemma telescopeBound indu
"" lemma telescopeBound

""; F/R
"" lemma lessNeq(F) helper
"" lemma lessNeq(F)
"" lemma lessNeq(R)

""; SUP
"" lemma intervalSize base
"" lemma intervalSize indu
"" lemma intervalSize
"" lemma XSlessUS
"" lemma USdecreasing(+1)
"" lemma closeUS
"" lemma closeUS(n+1)

""; PRAEXXDIKATLOGIK
"" pred lemma allNegated(Imply)
"" pred lemma existNegated(Imply)
"" pred lemma intro exist helper
"" pred lemma intro exist
"" pred lemma exist mp
"" pred lemma exist mp2
"" pred lemma 2exist mp
"" pred lemma 2exist mp2
"" pred lemma EAE mp
"" pred lemma addAll
"" pred lemma addExist helper1
"" pred lemma addExist helper2
"" pred lemma addExist
"" pred lemma addExist(SimpleAnt)
"" pred lemma addExist(Simple)
"" pred lemma addEAE
"" pred lemma AEAnegated
"" pred lemma EEAnegated

""; REGELLEMMAER
"" lemma induction
"" lemma leqAntisymmetry
"" lemma leqTransitivity
"" lemma leqAddition
"" lemma leqMultiplication
"" lemma reciprocal
"" lemma equality
"" lemma eqLeq
"" lemma eqAddition
"" lemma eqMultiplication

""; LEQ
"" lemma leqMultiplicationLeft
"" lemma leqLessEq
"" lemma lessLeq
"" lemma from leqGeq
"" lemma subLeqRight
"" lemma subLeqLeft
"" lemma leqPlus1
"" lemma positiveToRight(Leq)
"" lemma positiveToRight(Leq)(1 term)
"" lemma negativeToRight(Leq)
"" lemma positiveToLeft(Leq)
"" lemma negativeToLeft(Leq)
"" lemma negativeToLeft(Leq)(1 term)
"" lemma leqAdditionLeft
"" lemma leqSubtraction
"" lemma leqSubtractionLeft
"" lemma thirdGeq
"" lemma leqNegated
"" lemma addEquations(Leq)
"" lemma multiplyEquations(Leq)
"" lemma thirdGeqSeries

""; LESS
"" lemma leqNeqLess
"" lemma fromLess
"" lemma toLess
"" lemma fromNotLess
"" lemma toNotLess
"" lemma negativeLessPositive
"" lemma leqLessTransitivity
"" lemma lessLeqTransitivity
"" lemma lessTransitivity
"" lemma lessTotality
"" lemma subLessRight
"" lemma subLessLeft
"" lemma switchTerms(x<y-z)
"" lemma switchTerms(x-y<z)
"" lemma lessAddition
"" lemma lessAdditionLeft
"" lemma lessMultiplication
"" lemma lessMultiplicationLeft
"" lemma lessDivision
"" lemma positiveToRight(Less)
"" lemma positiveToLeft(Less)
"" lemma negativeToLeft(Less)
"" lemma negativeToRight(Less)
"" lemma addEquations(Less)
"" lemma addEquations(LeqLess)
"" lemma reciprocalToLeft(Less)
"" lemma lessNegated
"" lemma positiveNonzero
"" lemma positiveNegated
"" lemma nonpositiveNegated
"" lemma negativeNegated
"" lemma nonnegativeNegated
"" lemma positiveHalved
"" lemma positiveInverted

""; NUMERICAL
"" lemma nonnegativeNumerical
"" lemma negativeNumerical
"" lemma positiveNumerical
"" lemma nonpositiveNumerical
"" lemma |0|=0
"" lemma 0<=|x|
"" lemma x<=|x|
"" lemma fromPositiveNumerical
"" lemma sameNumerical
"" lemma signNumerical(+)
"" lemma signNumerical
"" lemma toNumericalLess
"" lemma fromNumericalGreater
"" lemma numericalDifference
"" lemma numericalDifferenceLess helper
"" lemma numericalDifferenceLess
"" lemma splitNumericalSumHelper
"" lemma splitNumericalSum(++)
"" lemma splitNumericalSum(--)
"" lemma splitNumericalSum(+-, smallNegative)
"" lemma splitNumericalSum(+-, bigNegative)
"" lemma splitNumericalSum(+-)
"" lemma splitNumericalSum(-+)
"" lemma splitNumericalSum
"" lemma splitNumericalProduct(++)
"" lemma splitNumericalProduct(+-)
"" lemma splitNumericalProduct
"" lemma insertMiddleTerm(Numerical)
"" lemma insertTwoMiddleTerms(Numerical)

""; PLUS
"" lemma three2twoTerms
"" lemma three2threeTerms

""; TIMES
"" lemma three2twoFactors
"" lemma three2threeFactors
"" lemma times(-1)
"" lemma times(-1)Left

""; MAX
"" lemma leqMax1
"" lemma leqMax2
"" lemma lessThanMax

""; REGNESTYKKER
"" lemma x+y=zBackwards
"" lemma x*y=zBackwards
"" lemma x=x+(y-y)
"" lemma x=x+y-y
"" lemma x=x*y*(1/y)
"" lemma insertMiddleTerm(Sum)
"" lemma insertTwoMiddleTerms(Sum)
"" lemma insertMiddleTerm(Difference)
"" lemma x*0+x=x
"" lemma x*0=0
"" lemma nonnegativeFactors
"" lemma nonzeroFactors
"" lemma positiveFactors
"" lemma plusTimesMinus
"" lemma minusTimesMinus
"" lemma (-1)*(-1)+(-1)*1=0
"" lemma (-1)*(-1)=1
"" lemma 0<1Helper
"" lemma 0<1
"" lemma 0<2
"" lemma 0<3
"" lemma 0<1/2
"" lemma 0<1/3
"" lemma x+x=2*x 
"" lemma x+x+x=3*x
"" lemma (1/2)x+(1/2)x=x
"" lemma (1/3)x+(1/3)x+(1/3)x=x
"" lemma -x-y=-(x+y)
"" lemma -x*y=-(x*y)
"" lemma -0=0

""; F-AFDELINGEN
"" lemma sameFsymmetry
"" lemma sameFtransitivity
"" lemma f2R(Plus)
"" lemma f2R(Times)

""; LESS(R)
"" lemma <<TransitivityHelper(Q)
"" lemma <<Transitivity

""; NUMMER 1
"" lemma <<==Reflexivity

""; NUMMER 2
"" lemma <<==AntisymmetryHelper(Q)

""; NUMMER 4
"" lemma fromNot<f(Weak) helper
"" lemma fromNot<f(Weak)
"" lemma fromNot<f(Strong) helper2
"" lemma fromNot<f(Strong) helper
"" lemma fromNot<f(Strong)
"" lemma fromNotSameF(Strongest) helper2
"" lemma fromNotSameF(Strongest) helper
"" lemma fromNotSameF(Strongest)
"" lemma toLess(F) helper
"" lemma toLess(F)
"" lemma fromNot<<
"" lemma toLess(R)
"" lemma leqTotality(R)

""; NUMMER 5
"" lemma fromNotSameF(Weak)(Helper)
"" lemma fromNotSameF(Weak)
"" lemma fromNotLess(F)
"" lemma ==Addition
"" lemma ==AdditionLeft

""; NUMMER 6
"" lemma fpart-Bounded base
"" lemma fpart-Bounded indu helper
"" lemma fpart-Bounded indu
"" lemma fpart-Bounded
"" lemma f-Bounded helper
"" lemma f-Bounded
"" lemma sameFmultiplication helper
"" lemma sameFmultiplication
"" lemma eqMultiplication(R)
"" lemma eqMultiplicationLeft(R)
"" lemma x*0=0(F)
"" lemma x*0=0(R)
"" lemma lessMultiplication(F) helper2
"" lemma lessMultiplication(F) helper
"" lemma lessMultiplication(F) 
"" lemma lessMultiplication(R)
"" lemma leqMultiplication(R)

""; NUMMER 7
"" lemma plusAssociativity(F)
""; NUMMER 8
"" lemma plus0(F)
""; NUMMER 10
"" lemma plusCommutativity(F)

""; NUMMER 11
"" lemma timesAssociativity(F)

""; NUMMER 12
"" lemma times1f

""; NUMMER 13
"" lemma 2cauchy helper
"" lemma 2cauchy
"" lemma reciprocalF nonzero
"" lemma reciprocalFny nonzero
"" lemma eventually=f to sameF helper
"" lemma eventually=f to sameF
"" lemma fromNotSameF(Strong) helper2
"" lemma fromNotSameF(Strong) helper
"" lemma fromNotSameF(Strong)
"" lemma sameFreciprocal helper
"" lemma sameFreciprocal
"" lemma from!!==
"" lemma reciprocal(R)

""; NUMMER 14
"" lemma timesCommutativity(F)

""; NUMMER 15
"" lemma distribution(F)


""; USORTERET
"" lemma fromMax(1)
"" lemma fromMax(2)
"" prop lemma to negated and
"" lemma positiveToRight(Less)(1 term)
"" pred lemma (A~)to(~E)
"" lemma ==Transitivity4
"" lemma plus0Left(R)
"" lemma x=x+(y-y)(R)
"" lemma x=x+y-y(R)
"" lemma positiveToRight(Eq)(R)
"" lemma subtractEquations(R)
"" lemma neqAddition(R)
"" lemma eqAdditionLeft(R)
"" lemma three2twoTerms(R)
"" lemma positiveToRight(Less)(R)
"" lemma three2threeTerms(R)
"" lemma positiveToRight(Less)(1 term)(R)
"" lemma toLeq(Advanced)(R)
"" lemma leqNeqLess(R)
"" lemma subLeqLeft(R)
"" lemma leqLessTransitivity(R)
"" lemma negativeToLeft(Eq)(R)
"" lemma negativeToRight(Less)(R)
"" lemma !!==Symmetry
"" lemma negativeToRight(Eq)(R)
"" lemma negativeToRight(Eq)(1 term)(R)
"" lemma doubleMinus(R)
"" lemma uniqueNegative(R)
"" lemma subtractEquationsLeft(R)
"" lemma eqNegated(R)
"" lemma neqNegated(R)
"" lemma leqNegated(R)
"" lemma lessNegated(R)
"" lemma -0=0(R)
"" lemma negativeNegated(R)
"" lemma from leqGeq(R)
"" lemma subLeqRight(R)
"" lemma fromLess(R)
"" lemma nonnegativeNumerical(R)
"" lemma negativeNumerical(R)
"" lemma 0<=|x|(R)
"" lemma positiveNegated(R)
"" lemma addEquations(R)
"" lemma distributionOut(R)
"" lemma ==Transitivity5
"" lemma x*0+x=x(R)
"" lemma x*0=0(R)fff
"" lemma times(-1)(R)
"" lemma times(-1)Left(R)
"" lemma -x-y=-(x+y)(R)
"" lemma lessTotality(R)
"" lemma positiveNumerical(R)
"" lemma signNumerical(+)(R)
"" lemma sameNumerical(R)
"" lemma minusNegated(R)
"" lemma signNumerical(R)
"" lemma numericalDifference(R)
"" lemma x<=|x|(R)
"" lemma USlimitIsUpperBound helper
"" lemma USlimitIsUpperBound
"" lemma (-1)*(-1)+(-1)*1=0(R)
"" lemma (-1)*(-1)=1(R)
"" lemma 0<1Helper(R)
"" lemma 0<1(R)
"" lemma expZero exact(R)
"" lemma positiveBase(R) base
"" lemma three2twoFactors(R)
"" lemma x=x*y*(1/y)(R)
"" lemma neqMultiplication(R)
"" lemma lessTransitivity(R)
"" lemma 0<2(R)
"" lemma sameExp(R) base
"" lemma sameExp(R) indu
"" lemma sameExp(R) 
"" lemma subNeqLeft(R)
"" lemma subNeqRight(R)
"" lemma nonzeroFactors(R)
"" lemma nonnegativeFactors(R)
"" lemma positiveFactors(R)
"" lemma lessDivision(R)
"" lemma 0<1/2(R)
"" lemma positiveToRight(Eq)(1 term)(R)
"" lemma exp(+1)(R)
"" lemma positiveBase(R) indu
"" lemma positiveBase(R)
"" lemma -x*y=-(x*y)(R)
"" lemma positiveToLeft(Eq)(R)
"" lemma times1Left(R)
"" lemma ==Transitivity6
"" lemma x+x=2*x(R)
"" lemma (1/2)x+(1/2)x=x(R)
"" lemma distributionOut(Minus)(R)
"" lemma (1/2)(x+y)-x=(1/2)(y-x)(R)
"" lemma intervalSize(R) base
"" lemma lessMultiplicationLeft(R)
"" lemma negativeToLeft(Less)(R)
"" lemma negativeToLeft(Less)(1 term)(R)
"" lemma y-(1/2)(x+y)=(1/2)(y-x)(R)
"" lemma intervalSize(R) indu
"" lemma intervalSize(R)
"" lemma XSlessUS(R)
"" lemma USdecreasing(+1)(R)
"" lemma expUnbounded base
"" lemma expUnbounded indu
"" lemma expUnbounded
"" lemma 1<=x+1(N)
"" lemma nonzeroProduct(2)(R)
"" lemma positiveNonzero(R)
"" lemma nonreciprocalToRight(Eq)(1 term)(R)
"" lemma expNonzero base
"" lemma expNonzero indu
"" lemma expNonzero
"" lemma expNonzero(2)
"" lemma multiplyEquations(R)
"" lemma halfBase base
"" lemma halfBase indu
"" lemma halfBase
"" lemma three2threeFactors(R)
"" lemma x*y=zBackwards(R)
"" lemma positiveInverted(R)
"" lemma reciprocalToRight(Less)(R)
"" lemma reciprocalToRight(Less)(1 term)(R)
"" lemma nonreciprocalToLeft(Less)(R)
"" lemma 1<x*y(R)
"" lemma switchFactors(1/x<y)(R)
"" lemma smallHalving
"" lemma intervalSize(anyPositive)
"" lemma USdecreasing(+n) base
"" lemma USdecreasing(+n) indu
"" lemma USdecreasing(+n)
"" lemma USdecreasing
"" lemma leqAdditionLeft(R)
"" lemma toNotLess(R)
"" lemma limitOfUSIsLeq
"" lemma subtractEquations(Less)(R)
"" lemma subtractEquationsLeft(Less)(R)
"" lemma lessNegated(Negative)(R)
"" prop lemma from negated and (imply)
"" prop lemma remove double neg (consequent)
"" lemma fromNotUpperBound
"" lemma distributionOut
"" lemma distributionOutLeft
"" lemma distributionLeft
"" lemma leqNUB
"" lemma USlimitIsLeastUpperBound helper
"" lemma USlimitIsLeastUpperBound
"" lemma fromNotLess(R)
"" pred lemma exist mp3
"" lemma greaterPositive(N)
"" lemma ysFClose helper
"" lemma ysFClose
"" lemma ysFCauchy helper
"" lemma ysFCauchy
"" lemma cartProdIsRelation
"" lemma fromSubset
"" lemma subsetIsRelation
"" lemma toSeries
"" lemma fromSeries
"" lemma seriesSubsetCP
"" lemma valueType 
"" prop lemma remove or
"" lemma fromSingleton
"" lemma inPair(1)
"" lemma inPair(2)
"" lemma sameMember(2)
"" lemma toBinaryUnion(1)
"" lemma toBinaryUnion(2)
"" lemma fromOrderedPair(twoLevels)
"" lemma toCartProd helper
"" lemma toCartProd
"" lemma nonreciprocalToRight(Eq)
"" lemma nonreciprocalToLeft(Eq)(1 term)
"" lemma sameReciprocal
"" lemma CPseparationIsRelation
"" lemma orderedPairEquality
"" lemma reciprocalIsFunction
"" lemma reciprocalIsTotal
"" lemma reciprocalIsRationalSeries
"" lemma crsIsRelation
"" lemma crsIsFunction
"" lemma crsIsTotal
"" lemma crsIsSeries
"" lemma crsLookup
"" lemma 0f
"" lemma 1f
"" lemma toSingleton
"" lemma fromSameSingleton 
"" lemma singletonmembersEqual
"" lemma unequalsNotInSingleton
"" lemma nonsingletonmembersUnequal
"" lemma fromOrderedPair
"" lemma fromOrderedPair(1)
"" lemma fromOrderedPair(2)
"" lemma fromCartProd
"" lemma fromCartProd(1)
"" lemma fromCartProd(2)
"" lemma sameOrderedPair
"" lemma inSeries helper
"" lemma inSeries
"" lemma to=f subset helper
"" lemma to=f subset
"" lemma to=f
"" lemma productIsFunction
"" lemma productIsTotal
"" lemma productIsRationalSeries
"" lemma timesF
"" lemma -x+(1/2)x=-(1/2)x
"" lemma positiveTripled
"" lemma positiveDividedBy3
"" lemma |x-x|=0
"" lemma 1<2
"" lemma 1/3<2/3
"" lemma (1/3)x+(1/3)x=(2/3)x
"" lemma (2/3)x+(1/3)x=x
"" lemma -x+(2/3)x=-(1/3)x
"" lemma -(1/3)x-(1/3)x=-(2/3)x
"" lemma -x+(1/3)x=-(2/3)x
"" lemma preserveLessGreater
"" lemma closetolessIsLess
"" lemma subLessLeft(F)
"" lemma subLessLeft(R)
"" lemma closetogreaterIsGreater
"" lemma subLessRight(F)
"" lemma subLessRight(R)



PREASSOCIATIVE
""  tester1
""  tester2
""  tester3
""  tester4
""  tester5 
""  tester6

PREASSOCIATIVE
"base" " sub " end sub 
"check" " hide
PREASSOCIATIVE
"base" unicode start of text " end unicode text 
"check" macro indent "
PREASSOCIATIVE
"base" " apply " 
PREASSOCIATIVE ""; RR 
"kvanti" " ^ "
PREASSOCIATIVE ""; RR
"check" " suc 
"kvanti" R( " )
PREASSOCIATIVE ""; RR
"kvanti" eq-system of " modulo "
PREASSOCIATIVE
"kvanti" union " end union
PREASSOCIATIVE
"kvanti" zermelo singleton " end singleton
PREASSOCIATIVE ""; RR
"kvanti" zermelo pair " comma " end pair
PREASSOCIATIVE 
"kvanti" " is related to " under "
PREASSOCIATIVE ""; RR 
"base" " times "
"kvanti" " * "
PREASSOCIATIVE ""; RR
"base" " plus "
"kvanti" " + "
PREASSOCIATIVE ""; XX in0
"kvanti" " in0 "
PREASSOCIATIVE ""; RR, Lukkede udtryk
"kvanti" | " | 
PREASSOCIATIVE ""; RR
"kvanti" " = "
PREASSOCIATIVE 
"base" " term plus " end plus
POSTASSOCIATIVE 
"base" " raw pair "
POSTASSOCIATIVE 
"base" " comma "
PREASSOCIATIVE 
"base" " boolean equal "
"check" " equal "
"kvanti" " zermelo is "
PREASSOCIATIVE 
"base" not "
"kvanti" not0 "
PREASSOCIATIVE 
"base" " and "
"kvanti" " and0 "
PREASSOCIATIVE 
"base" " or "
POSTASSOCIATIVE ""; XX or0 er 'pre' i 'equivalence-relations'
"kvanti" " or0 "
PREASSOCIATIVE ""; RR
"check"   exist  " indeed "
"kvanti"  exist0 " indeed "
POSTASSOCIATIVE 
"base" " macro imply "
"check" " imply "
"kvanti" " iff "
PREASSOCIATIVE 
"kvanti" the set of ph in " such that " end set
POSTASSOCIATIVE 
"base" " guard "
PREASSOCIATIVE 
"base" " select " else " end select
PREASSOCIATIVE 
 "base" lambda " dot "
PREASSOCIATIVE 
 "check" " avoid "
PREASSOCIATIVE 
 "base" " init
PREASSOCIATIVE 
 "base" " at "
"check" " object modus ponens " 
POSTASSOCIATIVE 
"base" " infer "
PREASSOCIATIVE 
"base" all " indeed "
"check" for all terms " indeed "
POSTASSOCIATIVE 
"base" " rule plus "
POSTASSOCIATIVE 
"base" " cut "
PREASSOCIATIVE  
"base" " proves "
PREASSOCIATIVE 
"base" locally define " as " end line "
"check" block " line " end block "
POSTASSOCIATIVE 
"check" " alternative "
POSTASSOCIATIVE 
"base" " , "
PREASSOCIATIVE 
"base" " tab "
PREASSOCIATIVE 
"base" " row "
"check" " safe row "

BODY

"File page.tex
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\begin {document}

\floatplacement{figure}{h!}
\floatplacement{table}{h!}
\hyphenation{her-ud-over ek-si-stens-va-ri-ab-le an-dre dob-belt-im-pli-ka-ti-on ob-jekt-kvan-tor de-fi-ni-tions-lem-ma ens-be-tyd-en-de 
und-er-af-snit slut-ning-en inde-hol-der for-klar-ing-en si-de-be-ting-el-sen
des-uden be-vis-check-er-en}

"[ ragged right expansion ]"

SUPSUPSUPSUPSUPSUPSUPSUPSUPSUPSUPSUPSUPSUPSUP



"[ math in theory system Q lemma prop lemma remove or says
for all terms meta a indeed
meta a or0 meta a infer meta a 
end lemma end math ]"

"[ math system Q proof of prop lemma remove or reads
any term meta a end line
line ell a premise meta a or0 meta a end line
line ell b because 1rule repetition modus ponens ell a indeed not0 meta a imply meta a end line ""; XXREP or
line ell c because prop lemma auto imply indeed meta a imply meta a end line
because prop lemma from negations modus ponens ell c modus ponens ell b indeed meta a qed
end math ]"

"[ math in theory system Q lemma lemma leqTransitivity says
for all terms meta x comma meta y comma meta z indeed
meta x <= meta y infer meta y <= meta z infer meta x <= meta z 
end lemma end math ]"

"[ math system Q proof of lemma leqTransitivity reads
any term meta x comma meta y comma meta z end line
line ell a premise meta x <= meta y end line
line ell b premise meta y <= meta z end line
line ell c because axiom leqTransitivity indeed
  meta x <= meta y imply meta y <= meta z imply meta x <= meta z end line
because prop lemma mp2 modus ponens ell c modus ponens ell a modus ponens ell b indeed  
  meta x <= meta z qed
end math ]"


"[ math in theory system Q lemma lemma leqAntisymmetry says
for all terms meta x comma meta y indeed
meta x <= meta y infer meta y <= meta x infer meta x = meta y
end lemma end math ]"

"[ math system Q proof of lemma leqAntisymmetry reads
any term meta x comma meta y end line
line ell a premise meta x <= meta y end line
line ell b premise meta y <= meta x end line
line ell c because axiom leqAntisymmetry indeed
  meta x <= meta y imply meta y <= meta x imply meta x = meta y end line
because prop lemma mp2 modus ponens ell c modus ponens ell a modus ponens ell b indeed  
  meta x = meta y qed
end math ]"

"[ math in theory system Q lemma lemma leqAddition says
for all terms meta x comma meta y comma meta z indeed
meta x <= meta y infer 
meta x + meta z <= meta y + meta z
end lemma end math ]"

"[ math system Q proof of lemma leqAddition reads
any term meta x comma meta y comma meta z end line
line ell a premise meta x <= meta y end line
line ell b because axiom leqAddition indeed
  meta x <= meta y imply meta x + meta z <= meta y + meta z end line
because 1rule mp modus ponens ell b modus ponens ell a indeed  
  meta x + meta z <= meta y + meta z qed
end math ]"

"[ math in theory system Q lemma lemma leqMultiplication says
for all terms meta x comma meta y comma meta z indeed
0 <= meta z infer meta x <= meta y infer 
meta x * meta z <= meta y * meta z
end lemma end math ]"

"[ math system Q proof of lemma leqMultiplication reads
any term meta x comma meta y comma meta z end line
line ell b premise 0 <= meta z end line
line ell a premise meta x <= meta y end line
line ell c because axiom leqMultiplication indeed
 0 <= meta z imply  meta x <= meta y imply meta x * meta z <= meta y * meta z end line
because prop lemma  mp2 modus ponens ell c modus ponens ell b modus ponens ell a indeed  
  meta x * meta z <= meta y * meta z qed
end math ]"



"[  math in theory system Q lemma lemma reciprocal says
for all terms meta x indeed
meta x != 0 infer meta x * 1/ meta x = 1
end lemma end math ]"

"[ math system Q proof of lemma reciprocal reads
any term meta x end line
line ell a premise meta x != 0 end line
line ell b because axiom reciprocal indeed meta x != 0  imply meta x * 1/ meta x = 1 end line
because 1rule mp modus ponens ell b modus ponens ell a indeed meta x * 1/ meta x = 1 qed
end math ]"


"[  math in theory system Q lemma lemma eqLeq says
for all terms meta x comma meta y indeed
meta x = meta y infer meta x <= meta y 
end lemma end math ]"

"[ math system Q proof of lemma eqLeq reads
any term meta x comma meta y end line
line ell a premise meta x = meta y end line
line ell b because axiom eqLeq indeed meta x = meta y imply meta x <= meta y end line
because 1rule mp modus ponens ell b modus ponens ell a indeed meta x <= meta y qed
end math ]"

"[ math in theory system Q lemma lemma eqAddition says
for all terms meta x comma meta y comma meta z indeed
meta x = meta y infer 
meta x + meta z = meta y + meta z
end lemma end math ]"

"[ math system Q proof of lemma eqAddition reads
any term meta x comma meta y comma meta z end line
line ell a premise meta x = meta y end line
line ell b because axiom eqAddition indeed
  meta x = meta y imply meta x + meta z = meta y + meta z end line
because 1rule mp modus ponens ell b modus ponens ell a indeed  
  meta x + meta z = meta y + meta z qed
end math ]"

"[ math in theory system Q lemma lemma eqMultiplication says
for all terms meta x comma meta y comma meta z indeed
meta x = meta y infer 
meta x * meta z = meta y * meta z
end lemma end math ]"

"[ math system Q proof of lemma eqMultiplication reads
any term meta x comma meta y comma meta z end line
line ell a premise meta x = meta y end line
line ell c because axiom eqMultiplication indeed
  meta x = meta y imply meta x * meta z = meta y * meta z end line
because 1rule mp modus ponens ell c modus ponens ell a indeed  
  meta x * meta z = meta y * meta z qed
end math ]"


"[  math in theory system Q lemma lemma equality says
for all terms meta x comma meta y comma meta z indeed
meta x = meta y infer meta x = meta z infer meta y = meta z
end lemma end math ]"

"[ math system Q proof of lemma equality reads
any term meta x comma meta y comma meta z end line
line ell a premise meta x = meta y end line
line ell b premise meta x = meta z end line
line ell c because axiom equality indeed 
  meta x = meta y imply meta x = meta z imply meta y = meta z end line
because prop lemma mp2 modus ponens ell c modus ponens ell a modus ponens ell b indeed  
  meta y = meta z qed
end math ]"

"[  math in theory system Q lemma lemma eqReflexivity says
for all terms meta x indeed
meta x = meta x
end lemma end math ]"

"[ math system Q proof of lemma eqReflexivity reads
any term meta x end line
line ell a because axiom leqReflexivity indeed meta x <= meta x end line
because lemma leqAntisymmetry modus ponens ell a modus ponens ell a indeed  meta x = meta x qed
end math ]"

"[  math in theory system Q lemma lemma eqSymmetry says
for all terms meta x comma meta y indeed
meta x = meta y infer meta y = meta x
end lemma end math ]"

"[ math system Q proof of lemma eqSymmetry reads
any term meta x comma meta y end line
line ell a premise meta x = meta y end line
line ell b because lemma eqReflexivity indeed meta x = meta x end line
because lemma equality modus ponens ell a modus ponens ell b indeed  
  meta y = meta x qed
end math ]"

"[  math in theory system Q lemma lemma eqTransitivity says
for all terms meta x comma meta y comma meta z indeed
meta x = meta y infer meta y = meta z infer meta x = meta z
end lemma end math ]"

"[ math system Q proof of lemma eqTransitivity reads
any term meta x comma meta y comma meta z end line
line ell a premise meta x = meta y end line
line ell b premise meta y = meta z end line
line ell c because lemma eqSymmetry modus ponens ell a indeed
  meta y = meta x end line
because lemma equality modus ponens ell c modus ponens ell b indeed  meta x = meta z qed
end math ]"

"[  math in theory system Q lemma lemma eqTransitivity4 says
for all terms meta x comma meta y comma meta z comma meta u indeed
meta x = meta y infer meta y = meta z infer meta z = meta u infer
meta x = meta u 
end lemma end math ]"

"[ math system Q proof of lemma eqTransitivity4 reads
any term meta x comma meta y comma meta z comma meta u end line
line ell a premise meta x = meta y end line
line ell b premise meta y = meta z end line
line ell c premise meta z = meta u end line
line ell d because lemma eqTransitivity modus ponens ell a modus ponens ell b indeed
  meta x = meta z end line
 because lemma eqTransitivity modus ponens ell d modus ponens ell c indeed
  meta x = meta u qed
end math ]"


"[  math in theory system Q lemma lemma eqTransitivity5 says
for all terms meta x comma meta y comma meta z comma meta u comma meta v indeed
meta x = meta y infer meta y = meta z infer meta z = meta u infer
meta u = meta v infer meta x = meta v
end lemma end math ]"

"[ math system Q proof of lemma eqTransitivity5 reads
any term meta x comma meta y comma meta z comma meta u comma meta v end line
line ell a premise meta x = meta y end line
line ell b premise meta y = meta z end line
line ell c premise meta z = meta u end line
line ell d premise meta u = meta v end line
line ell e because lemma eqTransitivity4 modus ponens ell a modus ponens ell b modus ponens ell c indeed
  meta x = meta u end line
because lemma eqTransitivity modus ponens ell e modus ponens ell d indeed 
meta x = meta v qed
end math ]"

"[  math in theory system Q lemma lemma eqTransitivity6 says
for all terms meta x comma meta y comma meta z comma meta u comma meta v comma meta w indeed
meta x = meta y infer meta y = meta z infer meta z = meta u infer
meta u = meta v infer meta v = meta w infer meta x = meta w
end lemma end math ]"

"[ math system Q proof of lemma eqTransitivity6 reads
any term meta x comma meta y comma meta z comma meta u comma meta v comma meta w end line
line ell a premise meta x = meta y end line
line ell b premise meta y = meta z end line
line ell c premise meta z = meta u end line
line ell d premise meta u = meta v end line
line ell e premise meta v = meta w end line
line ell f because lemma eqTransitivity5 modus ponens ell a modus ponens ell b modus ponens ell c 
  modus ponens ell d indeed meta x = meta v end line
because lemma eqTransitivity modus ponens ell f modus ponens ell e indeed 
meta x = meta w qed
end math ]"



"[  math in theory system Q lemma lemma induction says
for all terms meta v1 comma meta a comma meta b comma meta c indeed
"";Natvar( meta v1 ) endorse
meta-sub meta b is meta a where meta v1 is 0 end sub endorse 
meta-sub meta c is meta a where meta v1 is meta v1 + 1 end sub endorse 

meta b infer
meta a imply meta c infer 
meta a 
end lemma end math ]"

"[ math system Q proof of lemma induction reads
any term meta v1 comma meta a comma meta b comma meta c end line
"";line ell a side condition Natvar( meta v1 ) end line
line ell b side condition meta-sub meta b is meta a where meta v1 is 0 end sub end line
line ell c side condition meta-sub meta c is meta a where meta v1 is meta v1 + 1 end sub end line
line ell d premise meta b end line
line ell e premise meta a imply meta c end line 

line ell f because 1rule gen modus ponens ell e indeed 
  for all meta v1 indeed parenthesis meta a imply meta c end parenthesis end line

line ell g because axiom induction ""; XX modus probans ell a 
  modus probans ell b modus probans ell c indeed
  meta b imply 
  for all meta v1 indeed parenthesis meta a imply meta c end parenthesis imply 
  for all meta v1 indeed meta a  end line


line ell h because prop lemma mp2  modus ponens ell g modus ponens ell d modus ponens ell f indeed
  for all meta v1 indeed meta a end line

because lemma a4 at meta v1 modus ponens ell h indeed meta a qed
end math ]"

"[ math in theory system Q lemma lemma toSeries says
for all terms meta fx comma meta sy indeed
for all object r1 indeed ( object r1 in0 meta fx imply 
  isOrderedPair( object r1 , N , meta sy )  ) infer
for all object f1 comma object f2 comma object f3 comma object f4 indeed
  ( (o object f1 , object f2 ) in0 meta fx imply (o object f3 , object f4 ) in0 meta fx imply
  object f1 = object f3 imply object f2 = object f4 ) infer
for all object s1 indeed 
  ( object s1 in0 N imply exist0 object s2 indeed (o object s1 , object s2 ) in0 meta fx ) infer
isSeries( meta fx , meta sy )
end lemma end math ]"

"[ math system Q proof of lemma toSeries reads
any term meta fx comma meta sy end line
line ell d premise for all object r1 indeed ( object r1 in0 meta fx imply 
  isOrderedPair( object r1 , N , meta sy ) ) end line
line ell b premise for all object f1 comma object f2 comma object f3 comma object f4 indeed
  ( (o object f1 , object f2 ) in0 meta fx imply (o object f3 , object f4 ) in0 meta fx imply
  object f1 = object f3 imply object f2 = object f4 ) end line
line ell c premise for all object s1 indeed 
  ( object s1 in0 N  imply exist0 object s2 indeed (o object s1 , object s2 ) in0 meta fx ) end line

line ell e because 1rule repetition modus ponens ell d indeed ""; XXREP isRelation
  isRelation( meta fx , N , meta sy ) end line

line ell f because prop lemma join conjuncts modus ponens ell e modus ponens ell b indeed
  isRelation( meta fx , N , meta sy ) and0 
  for all object f1 comma object f2 comma object f3 comma object f4 indeed
  ( (o object f1 , object f2 ) in0 meta fx imply (o object f3 , object f4 ) in0 meta fx imply
  object f1 = object f3 imply object f2 = object f4 ) end line
line ell g because 1rule repetition modus ponens ell f indeed ""; XXREP isFunction
  isFunction( meta fx , N , meta sy ) end line

line ell h because prop lemma join conjuncts modus ponens ell g modus ponens ell c indeed
  isFunction( meta fx , N , meta sy ) and0 for all object s1 indeed 
  ( object s1 in0 N imply exist0 object s2 indeed (o object s1 , object s2 ) in0 meta fx ) end line

because 1rule repetition modus ponens ell h indeed isSeries( meta fx , meta sy ) qed ""; XXREP isSeries
end math ]"

"[ math in theory system Q lemma lemma fromSeries says
for all terms meta fx comma meta sy indeed
isSeries( meta fx , meta sy ) infer
( for all object r1 indeed ( object r1 in0 meta fx imply exist0 object op1 indeed exist0 object op2 indeed 
  object op1 in0 N and0 object op2 in0 meta sy and0 object r1 = (o object op1 , object op2 ) ) ) and0
( for all object f1 comma object f2 comma object f3 comma object f4 indeed
  ( (o object f1 , object f2 ) in0 meta fx imply (o object f3 , object f4 ) in0 meta fx imply
  object f1 = object f3 imply object f2 = object f4 ) ) and0
for all object s1 indeed 
  ( object s1 in0 N imply exist0 object s2 indeed (o object s1 , object s2 ) in0 meta fx )
end lemma end math ]"


"[ math system Q proof of lemma fromSeries reads
any term meta fx comma meta sy end line
line ell a premise isSeries( meta fx , meta sy ) end line
line ell b because 1rule repetition modus ponens ell a indeed ""; XXREP isSeries
  isFunction( meta fx , N , meta sy ) and0
  for all object s1 indeed 
    ( object s1 in0 N imply exist0 object s2 indeed (o object s1 , object s2 ) in0 meta fx ) end line
line ell c because 1rule repetition modus ponens ell b indeed ""; XXREP isFunction
  isRelation( meta fx , N , meta sy ) and0
  ( for all object f1 comma object f2 comma object f3 comma object f4 indeed
    ( (o object f1 , object f2 ) in0 meta fx imply (o object f3 , object f4 ) in0 meta fx imply
    object f1 = object f3 imply object f2 = object f4 ) ) and0
  for all object s1 indeed 
    ( object s1 in0 N imply exist0 object s2 indeed (o object s1 , object s2 ) in0 meta fx ) end line
line ell d because 1rule repetition modus ponens ell c indeed ""; XXREP isRelation
 ( for all object r1 indeed ( object r1 in0 meta fx imply isOrderedPair( object r1 , N , meta sy ) ) ) and0
  ( for all object f1 comma object f2 comma object f3 comma object f4 indeed
    ( (o object f1 , object f2 ) in0 meta fx imply (o object f3 , object f4 ) in0 meta fx imply
    object f1 = object f3 imply object f2 = object f4 ) ) and0
  for all object s1 indeed 
    ( object s1 in0 N imply exist0 object s2 indeed (o object s1 , object s2 ) in0 meta fx ) 
  end line
because 1rule repetition modus ponens ell d indeed ""; XXREP isOrderedPair
  ( for all object r1 indeed ( object r1 in0 meta fx imply exist0 object op1 indeed exist0 object op2 indeed 
    object op1 in0 N and0 object op2 in0 meta sy and0 object r1 = (o object op1 , object op2 ) ) ) and0
  ( for all object f1 comma object f2 comma object f3 comma object f4 indeed
    ( (o object f1 , object f2 ) in0 meta fx imply (o object f3 , object f4 ) in0 meta fx imply
    object f1 = object f3 imply object f2 = object f4 ) ) and0
  for all object s1 indeed 
    ( object s1 in0 N imply exist0 object s2 indeed (o object s1 , object s2 ) in0 meta fx ) qed
end math ]"

"[  math in theory system Q lemma pred lemma intro exist helper says
for all terms meta x comma meta v1 comma meta a comma meta b indeed
meta-sub not0 meta a is not0 meta b where meta v1 is meta x end sub endorse 
  for all meta v1 indeed not0 meta b imply not0 meta a
end lemma end math ]"

"[ math system Q proof of pred lemma intro exist helper reads
block any term meta x comma meta v1 comma meta a comma meta b end line
line ell big x side condition meta-sub not0 meta a is not0 meta b where meta v1 is meta x end sub end line
line ell b premise for all meta v1 indeed not0 meta b end line
because lemma a4 at meta x modus probans ell big x modus ponens ell b indeed not0 meta a end line
line ell big a end block

any term meta x comma meta v1 comma meta a comma meta b end line
because 1rule deduction modus ponens ell big a indeed
  meta-sub not0 meta a is not0 meta b where meta v1 is meta x end sub endorse 
  for all meta v1 indeed not0 meta b imply not0 meta a qed
end math ]"

"[  math in theory system Q lemma pred lemma intro exist says
for all terms meta x comma meta v1 comma meta a comma meta b indeed
meta-sub not0 meta a is not0 meta b where meta v1 is meta x end sub endorse 
meta a infer exist0 meta v1 indeed meta b
end lemma end math ]"

"[ math system Q proof of pred lemma intro exist reads
any term meta x comma meta v1 comma meta a comma meta b end line
line ell big x side condition meta-sub not0 meta a is not0 meta b where meta v1 is meta x end sub end line
line ell big y because pred lemma intro exist helper at meta x modus probans ell big x indeed
  for all meta v1 indeed not0 meta b imply not0 meta a end line

line ell a premise meta a end line
line ell b because prop lemma add double neg modus ponens ell a indeed not0 not0 meta a end line
line ell c because prop lemma mt modus ponens ell big y modus ponens ell b indeed
  not0 for all meta v1 indeed not0 meta b end line

because 1rule repetition modus ponens ell c indeed exist0 meta v1 indeed meta b qed ""; XXREP exist
end math ]"

"[  math in theory system Q lemma pred lemma exist mp says
for all terms meta v1 comma meta a comma meta b indeed
meta a imply meta b infer 
exist0 meta v1 indeed meta a infer meta b
end lemma end math ]"

"[ math system Q proof of pred lemma exist mp reads
block any term meta v1 comma meta a comma meta b end line
line ell b premise meta a imply meta b end line
line ell a premise exist0 meta v1 indeed meta a end line
line ell c premise not0 meta b end line
line ell d because prop lemma mt modus ponens ell b modus ponens ell c indeed not0 meta a end line
line ell e because 1rule gen modus ponens ell d indeed for all meta v1 indeed not0 meta a end line
line ell f because 1rule repetition modus ponens ell a indeed ""; XXREP exist
  not0 for all meta v1 indeed not0 meta a end line
because prop lemma from contradiction modus ponens ell e modus ponens ell f indeed not0 not0 meta b end line
line ell big a end block

any term meta v1 comma meta a comma meta b end line

line ell big b because 1rule deduction modus ponens ell big a indeed
  parenthesis meta a imply meta b end parenthesis imply
  exist0 meta v1 indeed meta a imply not0 meta b imply not0 not0 meta b end line

line ell a premise meta a imply meta b end line
line ell b premise exist0 meta v1 indeed meta a end line
line ell c because prop lemma mp2 modus ponens ell big b modus ponens ell a modus ponens ell b indeed
  not0 meta b imply not0 not0 meta b end line

line ell d because prop lemma imply negation modus ponens ell c indeed not0 not0 meta b end line

because prop lemma remove double neg modus ponens ell d indeed meta b qed
end math ]"



"[  math in theory system Q lemma pred lemma exist mp2 says
for all terms meta v1 comma meta v2 comma meta a comma meta b comma meta c indeed
meta a imply meta b imply meta c infer
exist0 meta v1 indeed meta a infer exist0 meta v2 indeed meta b infer meta c
end lemma end math ]"


"[ math system Q proof of pred lemma exist mp2 reads
any term meta v1 comma meta v2 comma meta a comma meta b comma meta c end line
line ell a premise meta a imply meta b imply meta c end line
line ell b premise exist0 meta v1 indeed meta a end line
line ell c premise exist0 meta v2 indeed meta b end line

line ell d because pred lemma exist mp modus ponens ell a modus ponens ell b indeed
  meta b imply meta c end line

because pred lemma exist mp modus ponens ell d modus ponens ell c indeed meta c qed
end math ]"

"[  math in theory system Q lemma pred lemma 2exist mp says
for all terms meta v1 comma meta v2 comma meta a comma meta b indeed
meta a imply meta b infer
exist0 meta v1 indeed exist0 meta v2 indeed meta a infer meta b
end lemma end math ]"

"[ math system Q proof of pred lemma 2exist mp reads

block any term meta v2 comma meta a comma meta b end line
line ell a premise meta a imply meta b end line
line ell b premise exist0 meta v2 indeed meta a end line
because pred lemma exist mp modus ponens ell a modus ponens ell b indeed meta b end line
line ell big a end block

any term meta v1 comma meta v2 comma meta a comma meta b end line


line ell a because 1rule deduction modus ponens ell big a indeed
  parenthesis meta a imply meta b end parenthesis imply exist0 meta v2 indeed meta a imply meta b end line

line ell b premise meta a imply meta b end line
line ell c premise exist0 meta v1 indeed exist0 meta v2 indeed meta a end line

line ell d because 1rule mp modus ponens ell a modus ponens ell b indeed
  exist0 meta v2 indeed meta a imply meta b end line
because pred lemma exist mp modus ponens ell d modus ponens ell c indeed meta b qed
end math ]"

"[  math in theory system Q lemma pred lemma 2exist mp2 says
for all terms meta v1 comma meta v2 comma meta v3 comma meta v4 
    comma meta a comma meta b comma meta c indeed
meta a imply meta b imply meta c infer
exist0 meta v1 indeed exist0 meta v2 indeed meta a infer
exist0 meta v3 indeed exist0 meta v4 indeed meta b infer meta c
end lemma end math ]"

"[ math system Q proof of pred lemma 2exist mp2 reads
any term meta v1 comma meta v2 comma meta v3 comma meta v4 comma meta a comma meta b comma meta c end line
line ell a premise meta a imply meta b imply meta c end line
line ell b premise exist0 meta v1 indeed exist0 meta v2 indeed meta a end line
line ell c premise exist0 meta v3 indeed exist0 meta v4 indeed meta b end line

line ell d because pred lemma 2exist mp modus ponens ell a modus ponens ell b indeed
  meta b imply meta c end line

because pred lemma 2exist mp modus ponens ell d modus ponens ell c indeed meta c qed
end math ]"


"[  math in theory system Q lemma lemma neqSymmetry says
for all terms meta x comma meta y indeed
meta x != meta y infer meta y != meta x 
end lemma end math ]"

"[ math system Q proof of lemma neqSymmetry reads
block any term meta x comma meta y end line
line ell a premise meta y = meta x end line
because lemma eqSymmetry modus ponens ell a indeed meta x = meta y end line
line ell b end block

any term meta x comma meta y end line
line ell c because 1rule deduction modus ponens ell b indeed meta y = meta x imply meta x = meta y end line
line ell d premise meta x != meta y end line
because prop lemma mt modus ponens ell c modus ponens ell d indeed meta y != meta x qed
end math ]"

"[  math in theory system Q lemma lemma subNeqLeft says
for all terms meta x comma meta y comma meta z indeed
meta x = meta y infer meta x != meta z infer meta y != meta z 
end lemma end math ]"

"[ math system Q proof of lemma subNeqLeft reads
any term meta x comma meta y comma meta z end line
line ell a premise meta x = meta y end line
line ell b premise meta x != meta z end line
line ell c because axiom equality indeed 
  meta y = meta x imply meta y = meta z imply meta x = meta z end line
line ell d because lemma eqSymmetry modus ponens ell a indeed
  meta y = meta x end line
line ell e because 1rule mp modus ponens ell c modus ponens ell d indeed
  meta y = meta z imply meta x = meta z end line
line ell f because prop lemma contrapositive modus ponens ell e indeed
  meta x != meta z imply meta y != meta z end line
because 1rule mp modus ponens ell f modus ponens ell b indeed meta y != meta z qed
end math ]"

"[ math in theory system Q lemma lemma inPair(1) says
for all terms meta sx comma meta sy indeed
meta sx in0 (p meta sx , meta sy )
end lemma end math ]"

"[ math system Q proof of lemma inPair(1) reads
any term meta sx comma meta sy end line
line ell a because lemma eqReflexivity indeed meta sx = meta sx end line
line ell b because prop lemma weaken or second modus ponens ell a indeed 
  meta sx = meta sx or0 meta sx = meta sy end line
because lemma formula2pair modus ponens ell b indeed meta sx in0 (p meta sx , meta sy ) qed
end math ]"

"[ math in theory system Q lemma lemma inPair(2) says
for all terms meta sx comma meta sy indeed
meta sy in0 (p meta sx , meta sy )
end lemma end math ]"

"[ math system Q proof of lemma inPair(2) reads
any term meta sx comma meta sy end line
line ell a because lemma eqReflexivity indeed meta sy = meta sy end line
line ell b because prop lemma weaken or first modus ponens ell a indeed
  meta sy = meta sx or0 meta sy = meta sy end line
because lemma formula2pair modus ponens ell b indeed meta sy in0 (p meta sx , meta sy ) qed
end math ]"



"[ math in theory system Q lemma lemma fromSingleton says
for all terms meta sx comma meta sy indeed
meta sx in0 (s meta sy ) infer meta sx = meta sy
end lemma end math ]"

"[ math system Q proof of lemma fromSingleton reads
any term meta sx comma meta sy end line
line ell a premise meta sx in0 (s meta sy ) end line
line ell b because 1rule repetition modus ponens ell a indeed 
  meta sx in0 (p meta sy , meta sy ) end line ""; XXREP (s )
line ell c because lemma pair2formula modus ponens ell b indeed
  meta sx = meta sy or0 meta sx = meta sy end line
because prop lemma remove or modus ponens ell c indeed meta sx = meta sy qed
end math ]"

"[ math in theory system Q lemma lemma toSingleton says
for all terms meta sx comma meta sy indeed
meta sx = meta sy infer meta sx in0 (s meta sy )
end lemma end math ]"

"[ math system Q proof of lemma toSingleton reads
any term meta sx comma meta sy end line
line ell a premise meta sx = meta sy end line
line ell b because prop lemma weaken or first modus ponens ell a indeed
  meta sx = meta sy or0 meta sx = meta sy end line
line ell c because lemma formula2pair modus ponens ell b indeed
  meta sx in0 (p meta sy , meta sy ) end line
because 1rule repetition modus ponens ell c indeed meta sx in0 (s meta sy ) qed ""; XXREP (s )
end math ]"

"[ math in theory system Q lemma lemma fromSameSingleton says
for all terms meta sx comma meta sy indeed
(s meta sx ) = (s meta sy ) infer meta sx = meta sy
end lemma end math ]"

"[ math system Q proof of lemma fromSameSingleton reads
any term meta sx comma meta sy end line
line ell a premise (s meta sx ) = (s meta sy ) end line

line ell b because lemma eqReflexivity indeed meta sx = meta sx end line
line ell c because lemma toSingleton modus ponens ell b indeed meta sx in0 (s meta sx ) end line
line ell d because lemma set equality nec condition(1) modus ponens ell a modus ponens ell c indeed
  meta sx in0 (s meta sy ) end line
because lemma fromSingleton modus ponens ell d indeed  meta sx = meta sy qed
end math ]"




"[ math in theory system Q lemma lemma singletonmembersEqual says
for all terms meta sx comma meta sy comma meta sz indeed
(p meta sx , meta sy ) = (s meta sz ) infer meta sx = meta sy
end lemma end math ]"

"[ math system Q proof of lemma singletonmembersEqual reads
any term meta sx comma meta sy comma meta sz end line
line ell a premise (p meta sx , meta sy ) = (s meta sz ) end line

line ell b because lemma inPair(1) indeed meta sx in0 (p meta sx , meta sy ) end line
line ell c because lemma set equality nec condition(1) modus ponens ell a modus ponens ell b indeed
  meta sx in0 (s meta sz ) end line
line ell d because lemma fromSingleton modus ponens ell c indeed meta sx = meta sz end line

line ell e because lemma inPair(2) indeed meta sy in0 (p meta sx , meta sy ) end line
line ell f because lemma set equality nec condition(1) modus ponens ell a modus ponens ell e indeed
  meta sy in0 (s meta sz ) end line
line ell g because lemma fromSingleton modus ponens ell f indeed meta sy = meta sz end line
line ell h because lemma eqSymmetry modus ponens ell g indeed meta sz = meta sy end line

because lemma eqTransitivity modus ponens ell d modus ponens ell h indeed meta sx = meta sy qed
end math ]"

"[ math in theory system Q lemma lemma unequalsNotInSingleton says
for all terms meta sx comma meta sy comma meta sz indeed
meta sx != meta sy infer (p meta sx , meta sy ) != (s meta sz )
end lemma end math ]"

"[ math system Q proof of lemma unequalsNotInSingleton reads
block any term meta sx comma meta sy comma meta sz end line
line ell a premise (p meta sx , meta sy ) = (s meta sz ) end line
because lemma singletonmembersEqual modus ponens ell a indeed meta sx = meta sy end line
line ell big a end block

any term meta sx comma meta sy comma meta sz end line

line ell big b because 1rule deduction modus ponens ell big a indeed
  (p meta sx , meta sy ) = (s meta sz ) imply meta sx = meta sy end line

line ell a premise meta sx != meta sy end line
because prop lemma mt modus ponens ell big b modus ponens ell a indeed 
 (p meta sx , meta sy ) != (s meta sz ) qed
end math ]"

"[ math in theory system Q lemma lemma nonsingletonmembersUnequal says
for all terms meta sx comma meta sy indeed
(p meta sx , meta sy ) != (s meta sx ) infer meta sx != meta sy
end lemma end math ]"

"[ math system Q proof of lemma nonsingletonmembersUnequal reads
block any term meta sx comma meta sy end line
line ell a premise meta sx = meta sy end line
line ell b because lemma eqReflexivity indeed meta sx = meta sx end line
line ell c because lemma same pair modus ponens ell b modus ponens ell a indeed
  (p meta sx , meta sx ) = (p meta sx , meta sy ) end line
line ell d because 1rule repetition modus ponens ell c indeed 
  (s meta sx ) = (p meta sx , meta sy ) end line ""; XXREP (s )
because lemma eqSymmetry modus ponens ell d indeed (p meta sx , meta sy ) = (s meta sx ) end line
line ell big a end block

any term meta sx comma meta sy end line

line ell big b because 1rule deduction modus ponens ell big a indeed
  meta sx = meta sy imply (p meta sx , meta sy ) = (s meta sx ) end line

line ell a premise (p meta sx , meta sy ) != (s meta sx ) end line

because prop lemma mt modus ponens ell big b modus ponens ell a indeed meta sx != meta sy qed
end math ]"

"[ math in theory system Q lemma lemma fromOrderedPair says
for all terms meta sx comma meta sx1 comma meta sy comma meta sy1 indeed
(o meta sx , meta sy ) = (o meta sx1 , meta sy1 ) infer meta sx = meta sx1 and0 meta sy = meta sy1
end lemma end math ]"

"[ math system Q proof of lemma fromOrderedPair reads
block any term meta sx comma meta sx1 comma meta sy comma meta sy1 end line
line ell a premise meta sx1 = meta sy1 end line
line ell b premise (o meta sx , meta sy ) = (o meta sx1 , meta sy1 ) end line
line ell c because 1rule repetition modus ponens ell b indeed ""; XXREP (o )
  (p (s meta sx ) , (p meta sx , meta sy ) ) = (p (s meta sx1 ) , (p meta sx1 , meta sy1 ) ) end line

line ell e because lemma eqReflexivity indeed meta sx1 = meta sx1 end line
line ell f because lemma same pair modus ponens ell e modus ponens ell a indeed
  (p meta sx1 , meta sx1 ) = (p meta sx1 , meta sy1 ) end line
line ell g because 1rule repetition modus ponens ell f indeed ""; XXREP (s )
  (s meta sx1 ) = (p meta sx1 , meta sy1 ) end line
line ell h because lemma eqReflexivity indeed (s meta sx1 ) = (s meta sx1 ) end line
line ell i because lemma same pair modus ponens ell h modus ponens ell g indeed
  (p (s meta sx1 ) , (s meta sx1 ) ) = (p (s meta sx1 ) , (p meta sx1 , meta sy1 ) ) end line 
line ell j because 1rule repetition modus ponens ell i indeed ""; XXREP (s )
  (s (s meta sx1 ) ) = (p (s meta sx1 ) , (p meta sx1 , meta sy1 ) ) end line 
line ell k because lemma eqSymmetry modus ponens ell j indeed
   (p (s meta sx1 ) , (p meta sx1 , meta sy1 ) ) = (s (s meta sx1 ) ) end line 

line ell l because lemma eqTransitivity modus ponens ell c modus ponens ell k indeed
  (p (s meta sx ) , (p meta sx , meta sy ) ) = (s (s meta sx1 ) ) end line
line ell m because lemma inPair(1) indeed 
  (s meta sx ) in0 (p (s meta sx ) , (p meta sx , meta sy ) ) end line
line ell n because lemma set equality nec condition(1) modus ponens ell l modus ponens ell m indeed
  (s meta sx ) in0 (s (s meta sx1 ) ) end line
line ell o because lemma fromSingleton modus ponens ell n indeed (s meta sx ) = (s meta sx1 ) end line
line ell p because lemma fromSameSingleton modus ponens ell o indeed meta sx = meta sx1 end line
  
line ell q because lemma eqSymmetry modus ponens ell o indeed (s meta sx1 ) = (s meta sx ) end line
line ell r because lemma same singleton modus ponens ell q indeed
  (s (s meta sx1 ) ) = (s (s meta sx ) ) end line
line ell s because lemma eqTransitivity modus ponens ell l modus ponens ell r indeed 
  (p (s meta sx ) , (p meta sx , meta sy ) ) = (s (s meta sx ) ) end line
line ell t because lemma inPair(2) indeed
  (p meta sx , meta sy ) in0 (p (s meta sx ) , (p meta sx , meta sy ) ) end line
line ell u because lemma set equality nec condition(1) modus ponens ell s modus ponens ell t indeed
  (p meta sx , meta sy ) in0 (s (s meta sx ) ) end line
line ell v because lemma fromSingleton modus ponens ell u indeed 
  (p meta sx , meta sy ) = (s meta sx ) end line
line ell w because lemma singletonmembersEqual modus ponens ell v indeed
  meta sx = meta sy end line
line ell x because lemma eqSymmetry modus ponens ell w indeed meta sy = meta sx end line
line ell y because lemma eqTransitivity4 modus ponens ell x modus ponens ell p modus ponens ell a indeed
  meta sy = meta sy1 end line
because prop lemma join conjuncts modus ponens ell p modus ponens ell y indeed
  meta sx = meta sx1 and0 meta sy = meta sy1 end line
line ell big a end block

block any term meta sx comma meta sx1 comma meta sy comma meta sy1 end line
line ell a premise meta sx1 != meta sy1 end line
line ell b premise (o meta sx , meta sy ) = (o meta sx1 , meta sy1 ) end line
line ell c because 1rule repetition modus ponens ell b indeed ""; XXREP (s )
  (p (s meta sx ) , (p meta sx , meta sy ) ) = (p (s meta sx1 ) , (p meta sx1 , meta sy1 ) ) end line
line ell e because lemma inPair(1) indeed 
  (s meta sx ) in0 (p (s meta sx ) , (p meta sx , meta sy ) ) end line
line ell f because lemma set equality nec condition(1) modus ponens ell c modus ponens ell e indeed
  (s meta sx ) in0 (p (s meta sx1 ) , (p meta sx1 , meta sy1 ) ) end line
line ell g because lemma pair2formula modus ponens ell f indeed
  (s meta sx ) = (s meta sx1 ) or0 (s meta sx ) = (p meta sx1 , meta sy1 ) end line

line ell h because lemma unequalsNotInSingleton modus ponens ell a indeed
  (p meta sx1 , meta sy1 ) != (s meta sx ) end line
line ell i because lemma neqSymmetry modus ponens ell h indeed
  (s meta sx ) != (p meta sx1 , meta sy1 ) end line
line ell j because prop lemma negate second disjunct modus ponens ell g modus ponens ell i indeed
  (s meta sx ) = (s meta sx1 ) end line
line ell k because lemma fromSameSingleton modus ponens ell j indeed meta sx = meta sx1 end line

line ell m because lemma inPair(2) indeed
  (p meta sx1 , meta sy1 ) in0 (p (s meta sx1 ) , (p meta sx1 , meta sy1 ) ) end line
line ell n because lemma set equality nec condition(2) modus ponens ell c modus ponens ell m indeed
  (p meta sx1 , meta sy1 ) in0 (p (s meta sx ) , (p meta sx , meta sy ) ) end line
line ell o because lemma pair2formula modus ponens ell n indeed
  (p meta sx1 , meta sy1 ) = (s meta sx ) or0 (p meta sx1 , meta sy1 ) = (p meta sx , meta sy ) end line
line ell q because prop lemma negate first disjunct modus ponens ell o modus ponens ell h indeed
  (p meta sx1 , meta sy1 ) = (p meta sx , meta sy ) end line
line ell r because lemma inPair(2) indeed meta sy in0 (p meta sx , meta sy ) end line
line ell s because lemma set equality nec condition(2) modus ponens ell q modus ponens ell r indeed
  meta sy in0 (p meta sx1 , meta sy1 ) end line
line ell t because lemma pair2formula modus ponens ell s indeed
  meta sy = meta sx1 or0 meta sy = meta sy1 end line

line ell u because lemma unequalsNotInSingleton modus ponens ell a indeed 
  (p meta sx1 , meta sy1 ) != (s meta sx ) end line
line ell v because lemma subNeqLeft modus ponens ell q modus ponens ell u indeed
  (p meta sx , meta sy ) != (s meta sx ) end line
line ell w because lemma nonsingletonmembersUnequal modus ponens ell v indeed 
  meta sx != meta sy end line
line ell x because lemma subNeqLeft modus ponens ell k modus ponens ell w indeed
  meta sx1 != meta sy end line
line ell y because lemma neqSymmetry modus ponens ell x indeed
  meta sy != meta sx1 end line
line ell z because prop lemma negate first disjunct modus ponens ell t modus ponens ell y indeed
  meta sy = meta sy1 end line
because prop lemma join conjuncts modus ponens ell k modus ponens ell z indeed
  meta sx = meta sx1 and0 meta sy = meta sy1 end line
line ell big b end block

any term meta sx comma meta sx1 comma meta sy comma meta sy1 end line

line ell big c because 1rule deduction modus ponens ell big a indeed 
  meta sx1 = meta sy1 imply (o meta sx , meta sy ) = (o meta sx1 , meta sy1 ) imply 
  meta sx = meta sx1 and0 meta sy = meta sy1 end line

line ell big d because 1rule deduction modus ponens ell big b indeed
  meta sx1 != meta sy1 imply (o meta sx , meta sy ) = (o meta sx1 , meta sy1 ) imply 
  meta sx = meta sx1 and0 meta sy = meta sy1 end line

line ell a premise (o meta sx , meta sy ) = (o meta sx1 , meta sy1 ) end line
line ell b because prop lemma from negations modus ponens ell big c modus ponens ell big d indeed
  (o meta sx , meta sy ) = (o meta sx1 , meta sy1 ) imply 
  meta sx = meta sx1 and0 meta sy = meta sy1 end line

because 1rule mp modus ponens ell b modus ponens ell a indeed 
  meta sx = meta sx1 and0 meta sy = meta sy1 qed
end math ]"

"[ math in theory system Q lemma lemma fromOrderedPair(1) says
for all terms meta sx comma meta sx1 comma meta sy comma meta sy1 indeed
(o meta sx , meta sy ) = (o meta sx1 , meta sy1 ) infer meta sx = meta sx1
end lemma end math ]"

"[ math system Q proof of lemma fromOrderedPair(1) reads
any term meta sx comma meta sx1 comma meta sy comma meta sy1 end line
line ell a premise (o meta sx , meta sy ) = (o meta sx1 , meta sy1 ) end line
line ell b because lemma fromOrderedPair modus ponens ell a indeed 
  meta sx = meta sx1 and0 meta sy = meta sy1 end line
because prop lemma first conjunct modus ponens ell b indeed meta sx = meta sx1 qed
end math ]"

"[ math in theory system Q lemma lemma fromOrderedPair(2) says
for all terms meta sx comma meta sx1 comma meta sy comma meta sy1 indeed
(o meta sx , meta sy ) = (o meta sx1 , meta sy1 ) infer meta sy = meta sy1
end lemma end math ]"

"[ math system Q proof of lemma fromOrderedPair(2) reads
any term meta sx comma meta sx1 comma meta sy comma meta sy1 end line
line ell a premise (o meta sx , meta sy ) = (o meta sx1 , meta sy1 ) end line
line ell b because lemma fromOrderedPair modus ponens ell a indeed 
  meta sx = meta sx1 and0 meta sy = meta sy1 end line
because prop lemma second conjunct modus ponens ell b indeed meta sy = meta sy1 qed
end math ]"


"[ math in theory system Q lemma lemma sameMember(2) says 
for all terms meta sx comma meta sy comma meta sz indeed
meta sx = meta sy infer meta sy in0 meta sz infer meta sx in0 meta sz
end lemma end math ]"

"[ math system Q proof of lemma sameMember(2) reads
any term meta sx comma meta sy comma meta sz end line
line ell a premise meta sx = meta sy end line
line ell b premise meta sy in0 meta sz end line
line ell c because lemma eqSymmetry modus ponens ell a indeed meta sy = meta sx end line
because lemma sameMember modus ponens ell c modus ponens ell b indeed meta sx in0 meta sz qed
end math ]"

"[ math in theory system Q lemma lemma toBinaryUnion(1) says
for all terms meta sx comma meta sy comma meta sz comma meta su indeed ""; XX ekstra variable
meta sx in0 meta sy infer meta sx in0 binaryUnion( meta sy , meta sz ) 
end lemma end math ]"

"[ math system Q proof of lemma toBinaryUnion(1) reads
any term meta sx comma meta sy comma meta sz comma meta su end line
line ell a premise meta sx in0 meta sy end line
line ell d because lemma inPair(1) indeed meta sy in0 (p meta sy , meta sz )  end line
line ell e because prop lemma join conjuncts modus ponens ell a modus ponens ell d indeed
  meta sx in0 meta sy and0 meta sy in0 (p meta sy , meta sz ) end line
line ell f because pred lemma intro exist at meta sy modus ponens ell e indeed
  exist0 meta su indeed  meta sx in0 meta su and0 meta su in0 (p meta sy ,  meta sz  ) end line
line ell g because lemma formula2union modus ponens ell f indeed
  meta sx in0 U( (p meta sy , meta sz ) ) end line
because 1rule repetition modus ponens ell g indeed 
  meta sx in0 binaryUnion( meta sy , meta sz ) qed ""; XXREP binaryUnion
end math ]"



"[ math in theory system Q lemma lemma toBinaryUnion(2) says
for all terms meta sx comma meta sy comma meta sz comma meta su indeed ""; XX ekstra variable
meta sx in0 meta sz infer meta sx in0 binaryUnion( meta sy , meta sz ) 
end lemma end math ]"


"[ math system Q proof of lemma toBinaryUnion(2) reads
any term meta sx comma meta sy comma meta sz comma meta su end line
line ell a premise meta sx in0 meta sz end line
line ell d because lemma inPair(2) indeed meta sz in0 (p meta sy , meta sz ) end line
line ell e because prop lemma join conjuncts modus ponens ell a modus ponens ell d indeed
  meta sx in0 meta sz and0 meta sz in0 (p meta sy , meta sz ) end line
line ell f because pred lemma intro exist at meta sz modus ponens ell e indeed
  exist0 meta su indeed  meta sx in0 meta su and0 meta su in0 (p meta sy ,  meta sz  ) end line
line ell g because lemma formula2union modus ponens ell f indeed
  meta sx in0 U( (p meta sy , meta sz ) ) end line
because 1rule repetition modus ponens ell g indeed 
  meta sx in0 binaryUnion( meta sy , meta sz ) qed ""; XXREP binaryUnion
end math ]"

"[ math in theory system Q lemma lemma fromOrderedPair(twoLevels) says
for all terms meta sx comma meta sy comma meta sz comma meta su indeed
meta sx in0 meta sy infer meta sy in0 (o meta sz , meta su ) infer 
meta sx = meta sz or0 meta sx = meta su
end lemma end math ]"

"[ math system Q proof of lemma fromOrderedPair(twoLevels) reads
any term meta sx comma meta sy comma meta sz comma meta su end line
line ell a premise meta sx in0 meta sy end line
line ell b premise meta sy in0 (o meta sz , meta su ) end line
line ell c because 1rule repetition modus ponens ell b indeed 
  meta sy in0 (p (s meta sz ) , (p meta sz , meta su ) ) end line ""; XXREP (o )
line ell d because lemma pair2formula modus ponens ell c indeed
  meta sy = (s meta sz ) or0 meta sy = (p meta sz , meta su ) end line

block any term meta sx comma meta sy comma meta sz comma meta su end line
line ell b premise meta sy = (s meta sz ) end line
line ell a premise meta sx in0 meta sy end line
line ell c because lemma set equality nec condition(1) modus ponens ell b modus ponens ell a indeed
  meta sx in0 (s meta sz ) end line
line ell d because lemma fromSingleton modus ponens ell c indeed meta sx = meta sz end line
because prop lemma weaken or second modus ponens ell d indeed
  meta sx = meta sz or0 meta sx = meta su end line
line ell big a end block

block any term meta sx comma meta sy comma meta sz comma meta su end line
line ell b premise meta sy = (p meta sz , meta su ) end line
line ell a premise meta sx in0 meta sy end line
line ell c because lemma set equality nec condition(1) modus ponens ell b modus ponens ell a indeed
  meta sx in0 (p meta sz , meta su ) end line
because lemma pair2formula modus ponens ell c indeed 
  meta sx = meta sz or0 meta sx = meta su end line
line ell big b end block

line ell e because 1rule deduction modus ponens ell big a indeed
  meta sy = (s meta sz ) imply meta sx in0 meta sy imply meta sx = meta sz or0 meta sx = meta su end line
line ell f because 1rule deduction modus ponens ell big b indeed
  meta sy = (p meta sz , meta su ) imply meta sx in0 meta sy imply 
  meta sx = meta sz or0 meta sx = meta su end line
line ell g because prop lemma from disjuncts modus ponens ell d modus ponens ell e modus ponens ell f indeed
  meta sx in0 meta sy imply meta sx = meta sz or0 meta sx = meta su end line

because 1rule mp modus ponens ell g modus ponens ell a indeed meta sx = meta sz or0 meta sx = meta su qed
end math ]"




"[ math in theory system Q lemma lemma cartProdIsRelation says
for all terms meta sx comma meta sy indeed
isRelation( cartProd( meta sx , meta sy ) , meta sx , meta sy ) 
end lemma end math ]"

"[ math system Q proof of lemma cartProdIsRelation reads
block any term meta sx comma meta sy end line
line ell a premise object r1 in0 cartProd( meta sx , meta sy ) end line
line ell b because lemma separation2formula modus ponens ell a indeed
  object r1 in0 P( P( binaryUnion( meta sx , meta sy ) ) ) and0 
  isOrderedPair( object r1 , meta sx , meta sy ) end line
because prop lemma second conjunct modus ponens ell b indeed 
  isOrderedPair( object r1 , meta sx , meta sy ) end line
line ell big a end block

any term meta sx comma meta sy end line

line ell a because 1rule deduction modus ponens ell big a indeed
  object r1 in0 cartProd( meta sx , meta sy ) imply
  isOrderedPair( object r1 , meta sx , meta sy ) end line
line ell b because 1rule gen modus ponens ell a indeed
  for all object r1 indeed ( object r1 in0 cartProd( meta sx , meta sy ) imply
  isOrderedPair( object r1 , meta sx , meta sy ) ) end line

because 1rule repetition modus ponens ell b indeed 
  isRelation( cartProd( meta sx , meta sy ) , meta sx , meta sy ) qed ""; XXREP isRelation
end math ]"

"[ math in theory system Q lemma lemma fromSubset says
for all terms meta sx comma meta sy comma meta sz indeed
isSubset( meta sx , meta sy ) infer meta sz in0 meta sx infer meta sz in0 meta sy
end lemma end math ]"

"[ math system Q proof of lemma fromSubset reads
any term meta sx comma meta sy comma meta sz end line
line ell a premise isSubset( meta sx , meta sy ) end line
line ell b premise meta sz in0 meta sx end line
line ell c because 1rule repetition modus ponens ell a indeed ""; XXREPm isSubset
  for all object s1 indeed ( object s1 in0 meta sx imply object s1 in0 meta sy ) end line
line ell d because lemma a4 at meta sz modus ponens ell c indeed
  meta sz in0 meta sx imply meta sz in0 meta sy end line
because 1rule mp modus ponens ell d modus ponens ell b indeed meta sz in0 meta sy qed
end math ]"

"[ math in theory system Q lemma lemma subsetIsRelation says
for all terms meta sx comma meta sy comma meta sz comma meta su indeed
isRelation( meta sx , meta sz , meta su ) infer isSubset( meta sy , meta sx ) infer
isRelation( meta sy , meta sz , meta su ) 
end lemma end math ]"

"[ math system Q proof of lemma subsetIsRelation reads
block any term meta sx comma meta sy comma meta sz comma meta su end line
line ell a premise isRelation( meta sx , meta sz , meta su ) end line
line ell b premise isSubset( meta sy , meta sx ) end line
line ell c premise object r1 in0 meta sy end line

line ell d because 1rule repetition modus ponens ell a indeed ""; XXREP isRelation
  for all object r1 indeed ( object r1 in0 meta sx imply isOrderedPair( object r1 , meta sz , meta su ) )
  end line
line ell e because lemma a4 at object r1 modus ponens ell d indeed
  object r1 in0 meta sx imply isOrderedPair( object r1 , meta sz , meta su ) end line
line ell f because lemma fromSubset modus ponens ell b modus ponens ell c indeed 
  object r1 in0 meta sx end line
because 1rule mp modus ponens ell e modus ponens ell f indeed 
  isOrderedPair( object r1 , meta sz , meta su ) end line
line ell big a end block

any term meta sx comma meta sy comma meta sz comma meta su end line

line ell big b because 1rule deduction modus ponens ell big a indeed
  isRelation( meta sx , meta sz , meta su ) imply
  isSubset( meta sy , meta sx ) imply
  object r1 in0 meta sy imply
  isOrderedPair( object r1 , meta sz , meta su ) end line
line ell a premise isRelation( meta sx , meta sz , meta su ) end line
line ell b premise isSubset( meta sy , meta sx ) end line

line ell c because prop lemma mp2 modus ponens ell big b modus ponens ell a modus ponens ell b indeed
  object r1 in0 meta sy imply isOrderedPair( object r1 , meta sz , meta su ) end line
line ell d because 1rule gen modus ponens ell c indeed
  for all object r1 indeed ( object r1 in0 meta sy imply isOrderedPair( object r1 , meta sz , meta su ) ) end line
because 1rule repetition modus ponens ell d indeed 
  isRelation( meta sy , meta sz , meta su ) qed ""; XXREP isRelation
end math ]"

"[ math in theory system Q lemma lemma CPseparationIsRelation says
for all terms meta a comma meta sx comma meta sy indeed
isRelation( the set of ph in cartProd( meta sx , meta sy ) such that meta a end set , meta sx , meta sy )
end lemma end math ]"

"[ math system Q proof of lemma CPseparationIsRelation reads
block any term meta a comma meta sx comma meta sy end line
line ell a premise 
  object s1 in0 the set of ph in cartProd( meta sx , meta sy ) such that meta a end set end line
because lemma separation2formula(1) modus ponens ell a indeed
  object s1 in0 cartProd( meta sx , meta sy ) end line
line ell big a end block

any term meta a comma meta sx comma meta sy end line

line ell b because 1rule deduction modus ponens ell big a indeed for all object s1 indeed
  ( object s1 in0 the set of ph in cartProd( meta sx , meta sy ) such that meta a end set imply 
  object s1 in0 cartProd( meta sx , meta sy ) ) end line
line ell c because 1rule repetition modus ponens ell b indeed ""; XXREP isSubset
  isSubset( the set of ph in cartProd( meta sx , meta sy ) such that meta a end set , 
            cartProd( meta sx , meta sy ) ) end line
line ell d because lemma cartProdIsRelation indeed 
  isRelation( cartProd( meta sx , meta sy ) , meta sx  , meta sy ) end line
because lemma subsetIsRelation modus ponens ell d modus ponens ell c indeed isRelation( 
  the set of ph in cartProd( meta sx , meta sy ) such that meta a end set , meta sx , meta sy ) qed
end math ]"


"[ math in theory system Q lemma lemma toCartProd helper says
for all terms meta sx comma meta sx1 comma meta sy comma meta sy1 comma meta sz indeed
meta sx in0 meta sx1 infer meta sy in0 meta sy1 infer 
meta sz in0 (o meta sx , meta sy ) infer isSubset( meta sz , binaryUnion( meta sx1 , meta sy1 ) )
end lemma end math ]"

"[ math system Q proof of lemma toCartProd helper reads
block any term meta sx comma meta sx1 comma meta sy comma meta sy1 comma meta sz end line
line ell a premise meta sx in0 meta sx1 end line
line ell b premise meta sy in0 meta sy1 end line

line ell c premise meta sz in0 (o meta sx , meta sy ) end line
line ell d premise object s1 in0 meta sz end line
line ell e because lemma fromOrderedPair(twoLevels) modus ponens ell d modus ponens ell c indeed
  object s1 = meta sx or0 object s1 = meta sy end line

block any term meta sx comma meta sx1 comma meta sy1 end line
line ell b premise meta sx in0 meta sx1 end line
line ell a premise object s1 = meta sx end line
line ell c because lemma sameMember(2) modus ponens ell a modus ponens ell b indeed
  object s1 in0 meta sx1 end line
because lemma toBinaryUnion(1) modus ponens ell c indeed 
  object s1 in0 binaryUnion( meta sx1 , meta sy1 ) end line
line ell big a end block

block any term meta sx1 comma meta sy comma meta sy1 end line
line ell b premise meta sy in0 meta sy1 end line
line ell a premise object s1 = meta sy end line
line ell c because lemma sameMember(2) modus ponens ell a modus ponens ell b indeed
  object s1 in0 meta sy1 end line
because lemma toBinaryUnion(2) modus ponens ell c indeed 
  object s1 in0 binaryUnion( meta sx1 , meta sy1 ) end line
line ell big b end block

line ell f because 1rule deduction modus ponens ell big a indeed
  meta sx in0 meta sx1 imply object s1 = meta sx imply 
  object s1 in0 binaryUnion( meta sx1 , meta sy1 ) end line
line ell g because 1rule mp modus ponens ell f modus ponens ell a indeed  
  object s1 = meta sx imply object s1 in0 binaryUnion( meta sx1 , meta sy1 ) end line

line ell h because 1rule deduction modus ponens ell big b indeed
  meta sy in0 meta sy1 imply object s1 = meta sy imply 
  object s1 in0 binaryUnion( meta sx1 , meta sy1 ) end line
line ell i because 1rule mp modus ponens ell h modus ponens ell b indeed  
  object s1 = meta sy imply object s1 in0 binaryUnion( meta sx1 , meta sy1 ) end line
because prop lemma from disjuncts modus ponens ell e modus ponens ell g modus ponens ell i indeed
  object s1 in0 binaryUnion( meta sx1 , meta sy1 ) end line
line ell big c end block

any term meta sx comma meta sx1 comma meta sy comma meta sy1 comma meta sz end line

line ell big x because 1rule deduction modus ponens ell big c indeed
  meta sx in0 meta sx1 imply meta sy in0 meta sy1 imply meta sz in0 (o meta sx , meta sy ) imply
  object s1 in0 meta sz imply object s1 in0 binaryUnion( meta sx1 , meta sy1 ) end line

line ell a premise meta sx in0 meta sx1 end line
line ell b premise meta sy in0 meta sy1 end line
line ell c premise meta sz in0 (o meta sx , meta sy ) end line

line ell d because prop lemma mp3 modus ponens ell big x modus ponens ell a modus ponens ell b 
  modus ponens ell c indeed 
  object s1 in0 meta sz imply object s1 in0 binaryUnion( meta sx1 , meta sy1 ) end line
line ell e because 1rule gen modus ponens ell d indeed for all object s1 indeed (
  object s1 in0 meta sz imply object s1 in0 binaryUnion( meta sx1 , meta sy1 ) ) end line
because 1rule repetition modus ponens ell e indeed ""; XXREP isSubset
  isSubset( meta sz , binaryUnion( meta sx1 , meta sy1 ) ) qed
end math ]"



"[ math in theory system Q lemma lemma toCartProd says
for all terms meta sx comma meta sx1 comma meta sy comma meta sy1 indeed
meta sx in0 meta sx1 infer meta sy in0 meta sy1 infer 
(o meta sx , meta sy ) in0 cartProd( meta sx1 , meta sy1 )
end lemma end math ]"

"[ math system Q proof of lemma toCartProd reads
block any term meta sx comma meta sx1 comma meta sy comma meta sy1 end line
line ell a premise meta sx in0 meta sx1 end line
line ell b premise meta sy in0 meta sy1 end line
line ell c premise object s1 in0 (o meta sx , meta sy ) end line
line ell d because lemma toCartProd helper modus ponens ell a modus ponens ell b modus ponens ell c indeed
  isSubset( object s1 , binaryUnion( meta sx1 , meta sy1 ) ) end line
because lemma formula2power modus ponens ell d indeed
  object s1 in0 P( binaryUnion( meta sx1 , meta sy1 ) ) end line
line ell big a end block

any term meta sx comma meta sx1 comma meta sy comma meta sy1 end line

line ell big b because 1rule deduction modus ponens ell big a indeed
meta sx in0 meta sx1 imply meta sy in0 meta sy1 imply object s1 in0 (o meta sx , meta sy ) imply
  object s1 in0 P( binaryUnion( meta sx1 , meta sy1 ) ) end line

line ell a premise meta sx in0 meta sx1 end line
line ell b premise meta sy in0 meta sy1 end line

line ell d because prop lemma mp2 modus ponens ell big b modus ponens ell a modus ponens ell b indeed
  object s1 in0 (o meta sx , meta sy ) imply
  object s1 in0 P( binaryUnion( meta sx1 , meta sy1 ) ) end line
line ell e because 1rule gen modus ponens ell d indeed for all object s1 indeed (
  object s1 in0 (o meta sx , meta sy ) imply 
  object s1 in0 P( binaryUnion( meta sx1 , meta sy1 ) ) ) end line
line ell f because 1rule repetition modus ponens ell e indeed ""; XXREP isSubset
  isSubset( (o meta sx , meta sy ) , P( binaryUnion( meta sx1 , meta sy1 ) ) ) end line
line ell g because lemma formula2power modus ponens ell f indeed
  (o meta sx , meta sy ) in0 P( P( binaryUnion( meta sx1 , meta sy1 ) ) ) end line

line ell h because lemma eqReflexivity indeed (o meta sx , meta sy ) = (o meta sx , meta sy ) end line
line ell i because prop lemma join conjuncts modus ponens ell a modus ponens ell b indeed
  meta sx in0 meta sx1 and0 meta sy in0 meta sy1 end line
line ell j because prop lemma join conjuncts modus ponens ell i modus ponens ell h indeed
  meta sx in0 meta sx1 and0 meta sy in0 meta sy1 and0 
  (o meta sx , meta sy ) = (o meta sx , meta sy ) end line
line ell k because pred lemma intro exist at meta sy modus ponens ell j indeed exist0 object op2 indeed
  meta sx in0 meta sx1 and0 object op2 in0 meta sy1 and0 
  (o meta sx , meta sy ) = (o meta sx , object op2 ) end line


line ell l because pred lemma intro exist at meta sx modus ponens ell k indeed 
  exist0 object op1 indeed exist0 object op2 indeed
  object op1 in0 meta sx1 and0 object op2 in0 meta sy1 and0 
  (o meta sx , meta sy ) = (o object op1 , object op2 ) end line
line ell m because 1rule repetition modus ponens ell l indeed ""; XXREP isOrderedPair
  isOrderedPair( (o meta sx , meta sy ) , meta sx1 , meta sy1 ) end line

line ell n because lemma formula2separation modus ponens ell g modus ponens ell m indeed
  (o meta sx , meta sy ) in0 the set of ph in P( P( binaryUnion( meta sx1 , meta sy1 ) ) ) such that
     isOrderedPair( ph1 , meta sx1 , meta sy1 ) end set end line
because 1rule repetition modus ponens ell n indeed ""; XXREP cartProd
  (o meta sx , meta sy ) in0 cartProd( meta sx1 , meta sy1 ) qed
end math ]"



"[ math in theory system Q lemma lemma crsIsRelation says
for all terms meta x indeed
isRelation( constantRationalSeries( meta x ) , N , Q )
end lemma end math ]"

"[ math system Q proof of lemma crsIsRelation reads

block any term meta x end line
line ell a premise object s1 in0 constantRationalSeries( meta x ) end line
line ell b because 1rule repetition modus ponens ell a indeed ""; XXREP constantRationalSeries
  object s1 in0 the set of ph in cartProd( N , Q ) such that
    exist0 object crs1 indeed ph3 = (o object crs1 , meta x ) end set end line
line ell c because lemma separation2formula modus ponens ell b indeed
  object s1 in0 cartProd( N , Q ) and0 exist0 object crs1 indeed 
  object s1 = (o object crs1 , meta x ) end line
because prop lemma first conjunct modus ponens ell c indeed
  object s1 in0 cartProd( N , Q ) end line
line ell big a end block

any term meta x end line
line ell a because 1rule deduction modus ponens ell big a indeed
  object s1 in0 constantRationalSeries( meta x ) imply object s1 in0 cartProd( N , Q ) end line
line ell b because 1rule gen modus ponens ell a indeed for all object s1 indeed 
  ( object s1 in0 constantRationalSeries( meta x ) imply object s1 in0 cartProd( N , Q ) ) end line
line ell c because 1rule repetition modus ponens ell b indeed ""; XXREP isSubset
  isSubset( constantRationalSeries( meta x ) , cartProd( N , Q ) ) end line
line ell d because lemma cartProdIsRelation indeed isRelation( cartProd( N , Q ) , N , Q ) end line
because lemma subsetIsRelation modus ponens ell d modus ponens ell c indeed 
  isRelation( constantRationalSeries( meta x ) , N , Q ) qed
end math ]"



"[ math in theory system Q lemma lemma crsIsFunction says
for all terms meta x indeed
isFunction( constantRationalSeries( meta x ) , N , Q )
end lemma end math ]"

"[ math system Q proof of lemma crsIsFunction reads
block any term meta x end line ""; XX der er to anvendelser af crs1 - det stinker.
line ell a premise (o object f1 , object f2 ) = (o object crs1 , meta x ) end line
line ell b premise (o object f3 , object f4 ) = (o object crs1 , meta x ) end line

line ell c because lemma fromOrderedPair modus ponens ell a indeed
  object f1 = object crs1 and0 object f2 = meta x end line
line ell d because prop lemma second conjunct modus ponens ell c indeed object f2 = meta x end line

line ell e because lemma fromOrderedPair modus ponens ell b indeed
    object f3 = object crs1 and0 object f4 = meta x end line
line ell f because prop lemma second conjunct modus ponens ell e indeed object f4 = meta x end line
line ell g because lemma eqSymmetry modus ponens ell f indeed meta x = object f4 end line
because lemma eqTransitivity modus ponens ell d modus ponens ell g indeed object f2 = object f4 end line
line ell big a end block

block any term meta x end line
line ell big b because 1rule deduction modus ponens ell big a indeed
  (o object f1 , object f2 ) = (o object crs1 , meta x ) imply
  (o object f3 , object f4 ) = (o object crs1 , meta x ) imply
  object f2 = object f4 end line

line ell a premise (o object f1 , object f2 ) in0 constantRationalSeries( meta x ) end line
line ell b premise (o object f3 , object f4 ) in0 constantRationalSeries( meta x ) end line
line ell c premise object f1 = object f3 end line

line ell d because lemma separation2formula modus ponens ell a indeed
  (o object f1 , object f2 ) in0 cartProd( N , Q ) and0
  exist0 object crs1 indeed (o object f1 , object f2 ) = (o object crs1 , meta x ) end line
line ell e because prop lemma second conjunct modus ponens ell d indeed
  exist0 object crs1 indeed (o object f1 , object f2 ) = (o object crs1 , meta x ) end line

line ell f because lemma separation2formula modus ponens ell b indeed
  (o object f3 , object f4 ) in0 cartProd( N , Q ) and0
  exist0 object crs1 indeed (o object f3 , object f4 ) = (o object crs1 , meta x ) end line
line ell g because prop lemma second conjunct modus ponens ell f indeed
  exist0 object crs1 indeed (o object f3 , object f4 ) = (o object crs1 , meta x ) end line

because pred lemma exist mp2 modus ponens ell big b modus ponens ell e modus ponens ell g indeed
  object f2 = object f4 end line
line ell big c end block

any term meta x end line
line ell a because 1rule deduction modus ponens ell big c indeed
  for all object f1 comma object f2 comma object f3 comma object f4 indeed
  ( (o object f1 , object f2 ) in0 constantRationalSeries( meta x ) imply 
    (o object f3 , object f4 ) in0 constantRationalSeries( meta x ) imply
    object f1 = object f3 imply object f2 = object f4 ) end line
line ell b because lemma crsIsRelation indeed 
  isRelation( constantRationalSeries( meta x ) , N , Q ) end line

because prop lemma join conjuncts modus ponens ell b modus ponens ell a indeed 
  isFunction( constantRationalSeries( meta x ) , N , Q ) qed
end math ]"

"[ math in theory system Q lemma lemma crsIsTotal says
for all terms meta m comma meta x indeed
typeRational( meta x ) endorse
meta m in0 N infer (o meta m , meta x ) in0 constantRationalSeries( meta x )
end lemma end math ]"

"[ math system Q proof of lemma crsIsTotal reads
any term meta m comma meta x end line
line ell big a side condition typeRational( meta x ) end line
line ell a premise meta m in0 N end line
line ell b because axiom rationalType modus probans ell big a indeed meta x in0 Q end line
line ell c because lemma toCartProd modus ponens ell a modus ponens ell b indeed
  (o meta m , meta x ) in0 cartProd( N , Q ) end line
line ell d because lemma eqReflexivity indeed (o meta m , meta x ) = (o meta m , meta x ) end line
line ell e because pred lemma intro exist at meta m modus ponens ell d indeed 
  exist0 object crs1 indeed (o meta m , meta x ) = (o object crs1 , meta x ) end line
because lemma formula2separation modus ponens ell c modus ponens ell e indeed
  (o meta m , meta x ) in0 constantRationalSeries( meta x ) qed
end math ]"

"[ math in theory system Q lemma lemma crsIsSeries says
for all terms meta x indeed
isSeries( constantRationalSeries( meta x ) , Q )
end lemma end math ]"

"[ math system Q proof of lemma crsIsSeries reads
block any term meta x end line
line ell a premise object s1 in0 N end line
line ell b because lemma crsIsTotal modus ponens ell a indeed 
  (o object s1 , meta x ) in0 constantRationalSeries( meta x )  end line
because pred lemma intro exist at meta x modus ponens ell a indeed exist0 object s2 indeed 
  (o object s1 , object s2 ) in0 constantRationalSeries( meta x ) end line
line ell big a end block

any term meta x end line
line ell a because 1rule deduction modus ponens ell big a indeed 
  object s1 in0 N imply exist0 object s2 indeed 
  (o object s1 , object s2 ) in0 constantRationalSeries( meta x ) end line
line ell c because 1rule gen modus ponens ell a indeed for all object s1 indeed ( object s1 in0 N imply   
  exist0 object s2 indeed (o object s1 , object s2 ) in0 constantRationalSeries( meta x ) ) end line
line ell d because lemma crsIsFunction indeed 
  isFunction( constantRationalSeries( meta x ) , N , Q ) end line

because prop lemma join conjuncts modus ponens ell d modus ponens ell c indeed 
  isSeries( constantRationalSeries( meta x ) , Q ) qed
end math ]"


"[ math in theory system Q lemma lemma crsLookup says
for all terms meta m comma meta x indeed 
meta m in0 N infer 
[ constantRationalSeries( meta x ) ; meta m ] = meta x
end lemma end math ]"

"[ math system Q proof of lemma crsLookup reads
any term meta m comma meta x end line
line ell a premise meta m in0 N end line
line ell b because lemma crsIsSeries indeed isSeries( constantRationalSeries( meta x ) , Q ) end line
line ell c because lemma memberOfSeries modus ponens ell a modus ponens ell b indeed
  (o meta m , [ constantRationalSeries( meta x ) ; meta m ] ) in0 constantRationalSeries( meta x ) end line
line ell d because lemma crsIsTotal modus ponens ell a indeed
  (o meta m , meta x ) in0 constantRationalSeries( meta x ) end line
line ell e because lemma eqReflexivity indeed meta m = meta m end line
because lemma uniqueMember modus ponens ell b modus ponens ell c modus ponens ell d 
  modus ponens ell e indeed 
  [ constantRationalSeries( meta x ) ; meta m ] = meta x qed
end math ]"

"[ math in theory system Q lemma lemma 0f says
for all terms meta m indeed 
meta m in0 N infer 
[ 0f ; meta m ] = 0
end lemma end math ]"

"[ math system Q proof of lemma 0f reads
any term meta m end line
line ell a premise meta m in0 N end line
line ell b because lemma crsLookup modus ponens ell a indeed
  [ constantRationalSeries( 0 ) ; meta m ] = 0 end line
because 1rule repetition modus ponens ell b indeed [ 0f ; meta m ] = 0 qed ""; XXREP 0f
end math ]"


"[ math in theory system Q lemma lemma 1f says
for all terms meta m indeed 
meta m in0 N infer 
[ 1f ; meta m ] = 1
end lemma end math ]"



"[ math system Q proof of lemma 1f reads
any term meta m end line
line ell a premise meta m in0 N end line
line ell b because lemma crsLookup modus ponens ell a indeed
  [ constantRationalSeries( 1 ) ; meta m ] = 1 end line
because 1rule repetition modus ponens ell b indeed [ 1f ; meta m ] = 1 qed ""; XXREP 1f
end math ]"




-------(6.11.06, lemmaer fra kvanti, mod kronologien)






"[  math in theory system Q lemma lemma distributionOut says
for all terms meta x comma meta y comma meta z indeed
meta x * meta y + meta x * meta z = meta x * parenthesis meta y + meta z end parenthesis 
end lemma end math ]"

"[ math system Q proof of lemma distributionOut reads
any term meta x comma meta y comma meta z end line
line ell a because axiom distribution indeed 
  meta x * parenthesis meta y + meta z end parenthesis = meta x * meta y + meta x * meta z  end line
because lemma eqSymmetry modus ponens ell a indeed 
 meta x * meta y + meta x * meta z = meta x * parenthesis meta y + meta z end parenthesis qed
end math ]"

"[  math in theory system Q lemma lemma three2twoTerms says
for all terms meta x comma meta y comma meta z comma meta u indeed
meta y + meta z = meta u infer meta x + meta y + meta z = meta x + meta u
end lemma end math ]"

"[ math system Q proof of lemma three2twoTerms reads
any term meta x comma meta y comma meta z comma meta u end line
line ell a premise meta y + meta z = meta u end line
line ell b because lemma eqAdditionLeft modus ponens ell a indeed
  meta x + parenthesis meta y + meta z end parenthesis = meta x + meta u end line
line ell c because axiom plusAssociativity indeed
  meta x + meta y + meta z = meta x + parenthesis meta y + meta z end parenthesis end line
because lemma eqTransitivity modus ponens ell c modus ponens ell b indeed 
 meta x + meta y + meta z = meta x + meta u qed
end math ]"

"[  math in theory system Q lemma lemma three2threeTerms says
for all terms meta x comma meta y comma meta z indeed
meta x + meta y + meta z = meta x + meta z + meta y
end lemma end math ]"

"[ math system Q proof of lemma three2threeTerms reads
any term meta x comma meta y comma meta z end line
line ell a because axiom plusCommutativity indeed meta y + meta z = meta z + meta y end line
line ell b because lemma three2twoTerms modus ponens ell a indeed
  meta x + meta y + meta z = meta x + parenthesis meta z + meta y end parenthesis end line

line ell c because axiom plusAssociativity indeed
  meta x + meta z + meta y = meta x + parenthesis meta z + meta y end parenthesis end line 
line ell d because lemma eqSymmetry modus ponens ell c indeed
  meta x + parenthesis meta z + meta y end parenthesis = meta x + meta z + meta y end line

because lemma eqTransitivity modus ponens ell b modus ponens ell d indeed  
  meta x + meta y + meta z = meta x + meta z + meta y qed
end math ]"

"[  math in theory system Q lemma lemma three2twoFactors says
for all terms meta x comma meta y comma meta z comma meta u indeed
meta y * meta z = meta u infer meta x * meta y * meta z = meta x * meta u
end lemma end math ]"

"[ math system Q proof of lemma three2twoFactors reads
any term meta x comma meta y comma meta z comma meta u end line
line ell a premise meta y * meta z = meta u end line
line ell b because lemma eqMultiplicationLeft modus ponens ell a indeed
  meta x * parenthesis meta y * meta z end parenthesis = meta x * meta u end line
line ell c because axiom timesAssociativity indeed
  meta x * meta y * meta z = meta x * parenthesis meta y * meta z end parenthesis end line
because lemma eqTransitivity modus ponens ell c modus ponens ell b indeed 
 meta x * meta y * meta z = meta x * meta u qed
end math ]"


"[ math in theory system Q lemma lemma x=x+(y-y) says
for all terms meta x comma meta y indeed
meta x = meta x + parenthesis meta y - meta y end parenthesis
end lemma end math ]"

"[ math system Q proof of lemma x=x+(y-y) reads
any term meta x comma meta y end line
line ell a because axiom plus0 indeed meta x + 0 = meta x end line
line ell b because axiom negative indeed meta y - meta y = 0 end line
line ell c because lemma eqSymmetry modus ponens ell b indeed 0 = meta y - meta y end line
line ell d because lemma eqAdditionLeft modus ponens ell c indeed 
 meta x + 0 = meta x + parenthesis meta y - meta y end parenthesis end line
because lemma equality modus ponens ell a modus ponens ell d indeed  
  meta x = meta x + parenthesis meta y - meta y end parenthesis qed
end math ]"

"[ math in theory system Q lemma lemma x=x+y-y says
for all terms meta x comma meta y indeed
meta x = meta x + meta y - meta y 
end lemma end math ]"

"[ math system Q proof of lemma x=x+y-y reads
any term meta x comma meta y end line
line ell a because lemma x=x+(y-y) indeed
  meta x = meta x + parenthesis meta y - meta y end parenthesis end line

line ell b because axiom plusAssociativity indeed 
  meta x + meta y - meta y = meta x + parenthesis meta y - meta y end parenthesis end line
line ell c because lemma eqSymmetry modus ponens ell b indeed
  meta x + parenthesis meta y - meta y end parenthesis = meta x + meta y - meta y end line

because lemma eqTransitivity modus ponens ell a modus ponens ell c indeed 
  meta x = meta x + meta y - meta y qed
end math ]"

"[  math in theory system Q lemma lemma x=x*y*(1/y) says
for all terms meta x comma meta y indeed
meta y != 0 infer meta x = meta x * meta y * 1/ meta y
end lemma end math ]"

"[ math system Q proof of lemma x=x*y*(1/y) reads
any term meta x comma meta y end line
line ell a premise meta y != 0 end line
line ell b because axiom times1 indeed meta x * 1 = meta x end line

line ell c because lemma reciprocal modus ponens ell a indeed meta y * 1/ meta y = 1 end line
line ell d because lemma three2twoFactors modus ponens ell c indeed
  meta x * meta y * 1/ meta y = meta x * 1 end line

line ell e because lemma eqTransitivity modus ponens ell d modus ponens ell b indeed
  meta x * meta y * 1/ meta y = meta x end line
because lemma eqSymmetry modus ponens ell e indeed meta x = meta x * meta y * 1/ meta y qed
end math ]"

"[  math in theory system Q lemma lemma  x*0+x=x says
for all terms meta x indeed 
meta x * 0 + meta x = meta x 
end lemma end math ]"

"[ math system Q proof of lemma x*0+x=x reads
any term meta x end line
line ell big a because axiom times1 indeed meta x * 1 =  meta x end line
line ell a because lemma eqSymmetry modus ponens ell big a indeed  meta x = meta x * 1 end line
line ell b because lemma eqAdditionLeft modus ponens ell a indeed 
  meta x * 0 + meta x = meta x * 0 + meta x * 1 end line

line ell c because axiom distribution indeed 
  meta x * parenthesis 0 + 1 end parenthesis = meta x * 0 + meta x * 1 end line
line ell d because lemma eqSymmetry modus ponens ell c indeed 
  meta x * 0 + meta x * 1 = meta x * parenthesis 0 + 1 end parenthesis end line

line ell e   because lemma plus0Left indeed 0 + 1 = 1 end line

line ell f   because lemma eqMultiplicationLeft modus ponens ell e indeed 
  meta x * parenthesis 0 + 1 end parenthesis = meta x * 1 end line

because lemma eqTransitivity5 modus ponens ell b modus ponens ell d modus ponens ell f 
  modus ponens ell big a indeed  meta x * 0 + meta x = meta x qed
end math ]"

"[  math in theory system Q lemma lemma x*0=0 says
for all terms meta x indeed
meta x * 0 = 0
end lemma end math ]"

"[ math system Q proof of lemma x*0=0 reads
any term meta x end line
line ell a because lemma x=x+(y-y) indeed 
   meta x * 0 = meta x * 0 + parenthesis meta x - meta x end parenthesis  end line

line ell b because axiom plusAssociativity indeed 
  meta x * 0 + meta x - meta x = meta x * 0 + parenthesis meta x - meta x end parenthesis end line
line ell c because lemma eqSymmetry modus ponens ell b indeed
  meta x * 0 + parenthesis meta x - meta x end parenthesis = meta x * 0 + meta x - meta x end line

line ell d because lemma x*0+x=x indeed meta x * 0 + meta x = meta x end line
line ell e because lemma eqAddition modus ponens ell d indeed
  meta x * 0 + meta x - meta x = meta x - meta x end line

line ell f because axiom negative indeed meta x - meta x = 0 end line

because lemma eqTransitivity5 modus ponens ell a modus ponens ell c modus ponens ell e modus ponens ell f   
  indeed meta x * 0 = 0 qed
end math ]"

"[  math in theory system Q lemma lemma (-1)*(-1)+(-1)*1=0 says
(-1) * (-1) + (-1) * 1 = 0
end lemma end math ]"

"[ math system Q proof of lemma (-1)*(-1)+(-1)*1=0 reads
line ell a because lemma distributionOut indeed 
  (-1) * (-1) + (-1) * 1 = (-1) * parenthesis (-1) + 1 end parenthesis end line

line ell b because axiom negative  indeed 1 + (-1) = 0  end line
line ell c because axiom plusCommutativity indeed (-1) + 1 = 1 + (-1) end line
line ell d because lemma eqTransitivity modus ponens ell c modus ponens ell b indeed 
  (-1) + 1 = 0 end line
line ell e because lemma eqMultiplicationLeft modus ponens ell d indeed 
  (-1) * parenthesis (-1) + 1 end parenthesis = (-1) * 0 end line

line ell f because lemma x*0=0 indeed (-1) * 0 = 0 end line

because lemma eqTransitivity4 modus ponens ell a modus ponens ell e modus ponens ell f indeed  
(-1) * (-1) + (-1) * 1 = 0 qed
end math ]"

"[  math in theory system Q lemma lemma (-1)*(-1)=1 says
(-1) * (-1) = 1 
end lemma end math ]"


"[ math system Q proof of lemma (-1)*(-1)=1 reads
line ell a because lemma x=x+(y-y) indeed
  (-1) * (-1) = (-1) * (-1) + parenthesis 1 - 1 end parenthesis end line

line ell b because axiom times1 indeed (-1) * 1 = (-1) end line
line ell c because lemma eqSymmetry modus ponens ell b indeed (-1) = (-1) * 1 end line
line ell d because lemma eqAdditionLeft modus ponens ell c indeed 1 - 1 = 1 + (-1) * 1 end line
line ell e because lemma eqAdditionLeft modus ponens ell d indeed 
  (-1) * (-1) + parenthesis 1 - 1 end parenthesis = 
  (-1) * (-1) + parenthesis 1 + (-1) * 1 end parenthesis end line

line ell f because axiom plusCommutativity indeed 1 + (-1) * 1 = (-1) * 1 + 1 end line 
line ell g because lemma eqAdditionLeft modus ponens ell f indeed 
  (-1) * (-1) + parenthesis 1 + (-1) * 1 end parenthesis = 
  (-1) * (-1) + parenthesis (-1) * 1 + 1 end parenthesis end line

line ell big a because axiom plusAssociativity indeed
  (-1) * (-1) + (-1) * 1 + 1 = (-1) * (-1) + parenthesis (-1) * 1 + 1 end parenthesis  end line
line ell h because lemma eqSymmetry modus ponens ell big a indeed
  (-1) * (-1) + parenthesis (-1) * 1 + 1 end parenthesis = (-1) * (-1) + (-1) * 1 + 1 end line


line ell i because lemma (-1)*(-1)+(-1)*1=0 indeed
  (-1) * (-1) + (-1) * 1 = 0 end line
line ell j because lemma eqAddition modus ponens ell i indeed
  (-1) * (-1) + (-1) * 1 + 1 = 0 + 1 end line

line ell k because lemma plus0Left indeed 0 + 1 = 1 end line

line ell l because lemma eqTransitivity5 
modus ponens ell a modus ponens ell e modus ponens ell g modus ponens ell h indeed 
  (-1) * (-1) = (-1) * (-1) + (-1) * 1 + 1 end line
because lemma eqTransitivity4 modus ponens ell l modus ponens ell j modus ponens ell k indeed  
  (-1) * (-1) = 1 qed
end math ]"

"[  math in theory system Q lemma lemma subLeqRight says
for all terms meta x comma meta y comma meta z indeed
meta x = meta y infer meta z <= meta x infer meta z <= meta y
end lemma end math ]"

"[ math system Q proof of lemma subLeqRight reads
any term meta x comma meta y comma meta z end line
line ell a premise meta x  = meta y end line
line ell b premise meta z <= meta x end line
line ell c because lemma eqLeq modus ponens ell a indeed meta x <= meta y end line
because lemma leqTransitivity modus ponens ell b modus ponens ell c indeed 
  meta z <= meta y qed
end math ]"

"[  math in theory system Q lemma lemma subLeqLeft says
for all terms meta x comma meta y comma meta z indeed
meta x = meta y infer meta x <= meta z infer meta y <= meta z
end lemma end math ]"

"[ math system Q proof of lemma subLeqLeft reads
any term meta x comma meta y comma meta z end line
line ell a premise meta x = meta y end line 
line ell b premise meta x <= meta z end line
line ell c because lemma eqSymmetry modus ponens ell a indeed meta y = meta x end line

line ell d because lemma eqLeq modus ponens ell c indeed meta y <= meta x end line
because lemma leqTransitivity modus ponens ell d modus ponens ell b indeed meta y <= meta z qed
end math ]"


"[  math in theory system Q lemma lemma 0<1Helper says
1 <= 0 imply 0 <= 1 
end lemma end math ]"

"[ math system Q proof of lemma 0<1Helper reads
block
line ell a premise 1 <= 0 end line
line ell b because lemma leqAddition modus ponens ell a indeed 1 + (-1) <= 0 + (-1) end line
line ell c because axiom negative indeed 1 + (-1) = 0 end line 
line ell d because lemma subLeqLeft modus ponens ell c modus ponens ell b indeed 0 <= 0 + (-1) end line
line ell e because lemma plus0Left indeed 0 + (-1) = (-1) end line
line ell f because lemma subLeqRight modus ponens ell e modus ponens ell d indeed 0 <= (-1) end line
line ell g because  lemma leqMultiplication modus ponens ell f modus ponens ell f indeed 
  0 * (-1) <= (-1) * (-1) end line
line ell h because lemma x*0=0 indeed (-1) * 0 = 0 end line
line ell i because axiom timesCommutativity indeed 0 * (-1) = (-1) * 0 end line
line ell j because lemma eqTransitivity modus ponens ell i modus ponens ell h indeed 0 * (-1) = 0 end line
line ell k because lemma subLeqLeft modus ponens ell j modus ponens ell g indeed 0 <= (-1) * (-1) end line
line ell l because lemma (-1)*(-1)=1 indeed (-1) * (-1) = 1 end line


because lemma subLeqRight modus ponens ell l modus ponens ell k indeed 0 <= 1 end line
line ell m end block

because 1rule deduction modus ponens ell m indeed 1 <= 0 imply 0 <= 1 qed
end math ]"

"[  math in theory system Q lemma lemma 0<1 says
0 < 1 end lemma end math ]"

"[ math system Q proof of lemma 0<1 reads
line ell a because axiom leqTotality indeed 0 <= 1 or0 1 <= 0 end line
line ell b because prop lemma auto imply indeed 0 <= 1 imply 0 <= 1 end line
line ell c because lemma 0<1Helper indeed  1 <= 0 imply 0 <= 1 end line

line ell d because prop lemma from disjuncts modus ponens ell a modus ponens ell b modus ponens ell c indeed 0 <= 1  end line

line ell e because axiom 0not1 indeed 0 != 1 end line

because prop lemma join conjuncts modus ponens ell d modus ponens ell e indeed 0 < 1 qed
end math ]"

"[  math in theory system Q lemma lemma addEquations says
for all terms meta x comma meta y comma meta z comma meta u indeed
meta x = meta y infer meta z = meta u infer meta x + meta z = meta y + meta u
end lemma end math ]"

"[ math system Q proof of lemma addEquations reads
any term meta x comma meta y comma meta z comma meta u end line
line ell a premise meta x = meta y end line
line ell b premise meta z = meta u end line
line ell c because lemma eqAddition modus ponens ell a indeed meta x + meta z = meta y + meta z end line
line ell d because lemma eqAdditionLeft modus ponens ell b indeed meta y + meta z = meta y + meta u end line
because lemma eqTransitivity modus ponens ell c modus ponens ell d indeed 
  meta x + meta z = meta y + meta u qed end math ]"

"[  math in theory system Q lemma lemma subtractEquations says
for all terms meta x comma meta y comma meta z comma meta u indeed
meta x + meta z = meta y + meta u infer meta z = meta u infer meta x = meta y
end lemma end math ]"

"[ math system Q proof of lemma subtractEquations reads
any term meta x comma meta y comma meta z comma meta u end line
line ell b premise meta x + meta z = meta y + meta u end line
line ell a premise meta z = meta u end line

line ell c because lemma eqAddition modus ponens ell b indeed
  meta x + meta z - meta z = meta y + meta u - meta z end line

line ell d because lemma plus0Left indeed 0 + meta z = meta z end line
line ell e because lemma eqTransitivity modus ponens ell d modus ponens ell a indeed
  0 + meta z = meta u end line
line ell f because lemma positiveToRight(Eq) modus ponens ell e indeed 0 = meta u - meta z end line
line ell g because lemma eqSymmetry modus ponens ell f indeed meta u - meta z = 0 end line
line ell h because lemma eqAdditionLeft modus ponens ell g indeed 
  meta y + parenthesis meta u - meta z end parenthesis = meta y + 0 end line
line ell i because axiom plusAssociativity indeed
  meta y + meta u - meta z = meta y + parenthesis meta u - meta z end parenthesis end line
line ell j because axiom plus0 indeed meta y + 0 = meta y end line
line ell k because lemma eqTransitivity4 modus ponens ell i modus ponens ell h modus ponens ell j indeed
  meta y + meta u - meta z = meta y end line
line ell m because lemma x=x+y-y indeed meta x = meta x + meta z - meta z end line

 because lemma eqTransitivity4 modus ponens ell m modus ponens ell c modus ponens ell k indeed
  meta x = meta y qed
end math ]"

"[  math in theory system Q lemma lemma subtractEquationsLeft says
for all terms meta x comma meta y comma meta z comma meta u indeed
meta x + meta z = meta y + meta u infer meta x = meta y infer meta z = meta u
end lemma end math ]"

"[ math system Q proof of lemma subtractEquationsLeft reads
any term meta x comma meta y comma meta z comma meta u end line
line ell b premise meta x + meta z = meta y + meta u end line
line ell a premise meta x = meta y end line
line ell c because axiom plusCommutativity indeed meta z + meta x = meta x + meta z end line
line ell d because axiom plusCommutativity indeed meta y + meta u = meta u + meta y end line

line ell e because lemma eqTransitivity4 modus ponens ell c modus ponens ell b modus ponens ell d indeed
  meta z + meta x = meta u + meta y end line

because lemma subtractEquations modus ponens ell e modus ponens ell a indeed meta z = meta u qed
end math ]"

"[  math in theory system Q lemma lemma eqNegated says
for all terms meta x comma meta y indeed
meta x = meta y infer - meta x = - meta y 
end lemma end math ]"

"[ math system Q proof of lemma eqNegated reads
any term meta x comma meta y end line
line ell a premise meta x = meta y end line
line ell b because axiom negative indeed meta x - meta x = 0 end line
line ell c because axiom negative indeed meta y - meta y = 0 end line
line ell d because lemma eqSymmetry modus ponens ell c indeed 0 = meta y - meta y end line
line ell e because lemma eqTransitivity modus ponens ell b modus ponens ell d indeed
  meta x - meta x = meta y - meta y end line
because lemma subtractEquationsLeft modus ponens ell e modus ponens ell a indeed 
- meta x = - meta y qed
end math ]"

"[  math in theory system Q lemma lemma positiveToRight(Eq) says
for all terms meta x comma meta y comma meta z indeed
meta x + meta y = meta z infer meta x = meta z - meta y
end lemma end math ]"

"[ math system Q proof of lemma positiveToRight(Eq) reads
any term meta x comma meta y comma meta z end line
line ell a premise meta x + meta y = meta z end line
line ell b because lemma eqAddition modus ponens ell a indeed 
  meta x + meta y - meta y = meta z - meta y end line

line ell c because lemma x=x+y-y indeed
  meta x = meta x + meta y - meta y end line

because lemma eqTransitivity modus ponens ell c modus ponens ell b indeed  
  meta x = meta z - meta y qed
end math ]"

"[  math in theory system Q lemma lemma positiveToLeft(Eq)(1 term) says
for all terms meta x comma meta y indeed
meta x = meta y infer meta x - meta y = 0 
end lemma end math ]"

"[ math system Q proof of lemma positiveToLeft(Eq)(1 term) reads
any term meta x comma meta y end line
line ell a premise meta x = meta y end line
line ell b because lemma eqAddition modus ponens ell a indeed meta x - meta y = meta y - meta y end line
line ell c because axiom negative indeed meta y - meta y = 0 end line
because lemma eqTransitivity modus ponens ell b modus ponens ell c indeed meta x - meta y = 0 qed
end math ]"

"[  math in theory system Q lemma lemma positiveToRight(Leq) says
for all terms meta x comma meta y comma meta z indeed
meta x + meta y <= meta z infer meta x <= meta z - meta y
end lemma end math ]"

"[ math system Q proof of lemma positiveToRight(Leq) reads
any term meta x comma meta y comma meta z end line
line ell a premise meta x + meta y <= meta z end line
line ell b because lemma leqAddition modus ponens ell a indeed 
  meta x + meta y - meta y <= meta z - meta y end line

line ell c because lemma x=x+y-y indeed
  meta x = meta x + meta y - meta y end line
line ell d because lemma eqSymmetry modus ponens ell c indeed
  meta x + meta y - meta y = meta x end line

because lemma subLeqLeft modus ponens ell d modus ponens ell b indeed  
  meta x <= meta z - meta y qed
end math ]"



"[  math in theory system Q lemma lemma positiveToRight(Leq)(1 term) says 
for all terms meta y comma meta z indeed
meta y <= meta z infer 0 <= meta z - meta y 
end lemma end math ]"

"[ math system Q proof of lemma positiveToRight(Leq)(1 term) reads
any term meta y comma meta z end line
line ell a premise meta y <= meta z end line
line ell b because lemma plus0Left indeed 0 + meta y = meta y end line
line ell c because lemma eqSymmetry modus ponens ell b indeed meta y = 0 + meta y end line
line ell d because lemma subLeqLeft modus ponens ell c modus ponens ell a indeed 
  0 + meta y <= meta z end line

because lemma positiveToRight(Leq) modus ponens ell d indeed 0 <= meta z - meta y qed
end math ]"


"[  math in theory system Q lemma lemma negativeToLeft(Eq) says
for all terms meta x comma meta y comma meta z indeed
meta x = meta y - meta z infer meta x + meta z = meta y 
end lemma end math ]"

"[ math system Q proof of lemma negativeToLeft(Eq) reads
any term meta x comma meta y comma meta z end line
line ell a premise meta x = meta y - meta z end line
line ell b because lemma eqAddition modus ponens ell a indeed 
  meta x + meta z = meta y - meta z + meta z end line
line ell c because lemma three2threeTerms indeed
  meta y - meta z + meta z = meta y + meta z - meta z end line
line ell d because lemma x=x+y-y indeed
  meta y = meta y + meta z - meta z end line
line ell e because lemma eqSymmetry modus ponens ell d indeed
  meta y + meta z - meta z = meta y end line

because lemma eqTransitivity4 modus ponens ell b modus ponens ell c modus ponens ell e indeed  
  meta x + meta z = meta y qed
end math ]"

(*** NO EQUALITY ***)



"[  math in theory system Q lemma lemma lessNeq says
for all terms meta x comma meta y indeed 
meta x < meta y infer meta x != meta y
end lemma end math ]"

"[ math system Q proof of lemma lessNeq reads
any term meta x comma meta y end line
line ell a premise meta x < meta y end line
line ell b because 1rule repetition modus ponens ell a indeed ""; XXREP <
  meta x <= meta y and0 not0 parenthesis meta x = meta y end parenthesis end line
because prop lemma second conjunct modus ponens ell b indeed meta x != meta y qed
end math ]"



"[  math in theory system Q lemma lemma x+y=zBackwards says
for all terms meta x comma meta y comma meta z indeed
meta x + meta y = meta z infer meta z = meta y + meta x 
end lemma end math ]"

"[ math system Q proof of lemma x+y=zBackwards reads
any term meta x comma meta y comma meta z end line
line ell a premise meta x + meta y = meta z end line
line ell b because axiom plusCommutativity indeed meta x + meta y = meta y + meta x end line
because lemma equality modus ponens ell a indeed meta z = meta y + meta x qed
end math ]"

"[  math in theory system Q lemma lemma x*y=zBackwards says
for all terms meta x comma meta y comma meta z indeed
meta x * meta y = meta z infer meta z = meta y * meta x 
end lemma end math ]"

"[ math system Q proof of lemma x*y=zBackwards reads
any term meta x comma meta y comma meta z end line
line ell a premise meta x * meta y = meta z end line
line ell b because axiom timesCommutativity indeed meta x * meta y = meta y * meta x end line
because lemma equality modus ponens ell a indeed meta z = meta y * meta x qed
end math ]"



"[  math in theory system Q lemma lemma doubleMinus says
for all terms meta x indeed - - meta x = meta x 
end lemma end math ]"

"[ math system Q proof of lemma doubleMinus reads
any term meta x end line
line ell a because axiom negative indeed - meta x - - meta x = 0 end line
line ell b because lemma x+y=zBackwards modus ponens ell a indeed
  0 = - - meta x - meta x end line
line ell c because lemma negativeToLeft(Eq) modus ponens ell b indeed
  0 + meta x = - - meta x end line
line ell d because lemma plus0Left indeed 0 + meta x = meta x end line

because lemma equality modus ponens ell c modus ponens ell d indeed - - meta x = meta x qed
end math ]"



"[  math in theory system Q lemma lemma neqNegated says
for all terms meta x comma meta y indeed
meta x != meta y infer - meta x != - meta y
end lemma end math ]"

"[ math system Q proof of lemma neqNegated reads
block any term meta x comma meta y end line
line ell big a premise meta x != meta y end line
line ell big b premise - meta x = - meta y end line
line ell big c because lemma eqNegated modus ponens ell big b indeed - - meta x = - - meta y end line
line ell big d because lemma doubleMinus indeed - - meta x = meta x end line
line ell big e because lemma eqSymmetry modus ponens ell big d indeed meta x = - - meta x end line
line ell big f because lemma doubleMinus indeed - - meta y = meta y end line
line ell big g because lemma eqTransitivity4 modus ponens ell big e modus ponens ell big c modus ponens ell big f indeed meta x = meta y end line
because prop lemma from contradiction modus ponens ell big g modus ponens ell big a indeed
  - meta x != - meta y end line
line ell big h end block 

any term meta x comma meta y end line
line ell a because 1rule deduction modus ponens ell big h indeed
  meta x != meta y imply - meta x = - meta y imply not0 - meta x = - meta y end line
line ell b premise meta x != meta y end line
line ell c because 1rule mp modus ponens ell a modus ponens ell b indeed
  - meta x = - meta y imply not0 - meta x = - meta y end line
because prop lemma imply negation modus ponens ell c indeed not0 - meta x = - meta y qed
end math ]"

"[  math in theory system Q lemma lemma subNeqRight says
for all terms meta x comma meta y comma meta z indeed
meta x = meta y infer meta z != meta x infer meta z != meta y
end lemma end math ]"

"[ math system Q proof of lemma subNeqRight reads
any term meta x comma meta y comma meta z end line
line ell a premise meta x = meta y end line
line ell b premise meta z != meta x end line
line ell c because lemma neqSymmetry modus ponens ell b indeed meta x != meta z end line
line ell d because lemma subNeqLeft modus ponens ell a modus ponens ell c indeed
  meta y != meta z end line
because lemma neqSymmetry modus ponens ell d indeed meta z != meta y qed
end math ]"

% linie 4000

"[  math in theory system Q lemma lemma neqAddition says
for all terms meta x comma meta y comma meta z indeed
meta x != meta y infer meta x + meta z != meta y + meta z
end lemma end math ]"

"[ math system Q proof of lemma neqAddition reads
block any term meta x comma meta y comma meta z end line
line ell big a premise meta x != meta y end line
line ell big b premise meta x + meta z = meta y + meta z end line
line ell big c because lemma eqReflexivity indeed meta z = meta z end line
line ell big d because lemma subtractEquations modus ponens ell big b modus ponens ell big c indeed
  meta x = meta y end line
 because prop lemma from contradiction modus ponens ell big d modus ponens ell big a indeed
  meta x + meta z != meta y + meta z end line
line ell big e end block

any term meta x comma meta y comma meta z end line
line ell a because 1rule deduction modus ponens ell big e indeed
  meta x != meta y imply
  meta x + meta z = meta y + meta z imply meta x + meta z != meta y + meta z  end line
line ell b premise meta x != meta y end line
line ell c because 1rule mp modus ponens ell a modus ponens ell b indeed
  meta x + meta z = meta y + meta z imply meta x + meta z != meta y + meta z  end line
because prop lemma imply negation modus ponens ell c indeed 
  meta x + meta z != meta y + meta z qed
end math ]"


"[  math in theory system Q lemma lemma neqMultiplication says
for all terms meta x comma meta y comma meta z indeed
meta z != 0 infer meta x != meta y infer meta x * meta z != meta y * meta z
end lemma end math ]"

"[ math system Q proof of lemma neqMultiplication reads
block any term meta x comma meta y comma meta z end line
line ell a premise meta z != 0 end line
line ell b premise meta x != meta y end line

line ell c premise meta x * meta z = meta y * meta z end line

line ell d because lemma x=x*y*(1/y) modus ponens ell a indeed
  meta x = meta x * meta z * 1/ meta z end line

line ell e because lemma eqMultiplication modus ponens ell c indeed
  meta x * meta z * 1/ meta z = meta y * meta z * 1/ meta z end line

line ell f because lemma x=x*y*(1/y) modus ponens ell a indeed
  meta y = meta y * meta z * 1/ meta z end line
line ell g because lemma eqSymmetry modus ponens ell f indeed
  meta y * meta z * 1/ meta z = meta y end line

line ell h because lemma eqTransitivity4 modus ponens ell d modus ponens ell e modus ponens ell g indeed
  meta x = meta y end line

because prop lemma from contradiction modus ponens ell h modus ponens ell b indeed
  meta x * meta z != meta y * meta z end line
line ell i end block

any term meta x comma meta y comma meta z end line
line ell j because 1rule deduction modus ponens ell i indeed
  meta z != 0 imply meta x != meta y imply 
  meta x * meta z = meta y * meta z imply meta x * meta z != meta y * meta z end line

line ell k premise meta z != 0 end line
line ell l premise meta x != meta y end line

line ell m  because prop lemma mp2 modus ponens ell j modus ponens ell k modus ponens ell l indeed
  meta x * meta z = meta y * meta z imply meta x * meta z != meta y * meta z end line

because prop lemma imply negation modus ponens ell m indeed 
  meta x * meta z != meta y * meta z qed
end math ]"

(*** NEGATIVE ***)




"[  math in theory system Q lemma lemma uniqueNegative says
for all terms meta x comma meta y comma meta z indeed
meta x + meta y = 0 infer meta x + meta z = 0 infer meta y = meta z
end lemma end math ]"

"[ math system Q proof of lemma uniqueNegative reads
any term meta x comma meta y comma meta z end line
line ell a premise meta x + meta y = 0 end line
line ell b premise meta x + meta z = 0 end line

line ell c because axiom plusCommutativity indeed meta y + meta x = meta x + meta y end line
line ell d because lemma eqTransitivity modus ponens ell c modus ponens ell a indeed 
  meta y + meta x = 0 end line
line ell e because lemma positiveToRight(Eq) modus ponens ell d indeed meta y = 0 - meta x end line

line ell f because axiom plusCommutativity indeed meta z + meta x = meta x + meta z end line
line ell g because lemma eqTransitivity modus ponens ell f modus ponens ell b indeed 
  meta z + meta x = 0 end line
line ell h because lemma positiveToRight(Eq) modus ponens ell g indeed meta z = 0 - meta x end line
line ell i because lemma eqSymmetry modus ponens ell h indeed 0 - meta x = meta z end line

because lemma eqTransitivity modus ponens ell e modus ponens ell i indeed 
 meta y = meta z qed
end math ]"


"[  math in theory system Q lemma lemma fromLess says
for all terms meta x comma meta y indeed
meta x < meta y infer not0 meta y <= meta x 
end lemma end math ]"

"[ math system Q proof of lemma fromLess reads
block any term meta x comma meta y end line
line ell a premise meta y <= meta x end line
because lemma toNotLess modus ponens ell a indeed not0 meta x < meta y end line
line ell b end block 

any term meta x comma meta y end line
line ell c because 1rule deduction modus ponens ell b indeed meta y <= meta x imply not0 meta x < meta y end line
line ell d premise meta x < meta y end line
line ell e because prop lemma add double neg modus ponens ell d indeed 
  not0 not0 meta x < meta y end line
 because prop lemma mt modus ponens ell c modus ponens ell e indeed  
  not0 meta y <= meta x qed
end math ]"

"[  math in theory system Q lemma lemma toLess says
for all terms meta x comma meta y indeed
not0 meta x <= meta y infer meta y < meta x
end lemma end math ]"

"[ math system Q proof of lemma toLess reads 
block any term meta x comma meta y end line
line ell a premise not0 meta y < meta x end line
because lemma fromNotLess modus ponens ell a indeed meta x <= meta y end line
line ell b end block
any term meta x comma meta y end line
line ell c because 1rule deduction modus ponens ell b indeed 
  not0 meta y < meta x imply meta x <= meta y end line
line ell d premise not0 meta x <= meta y end line
because prop lemma negative mt modus ponens ell c modus ponens ell d indeed meta y < meta x qed
end math ]"



"[  math in theory system Q lemma lemma fromNotLess says
for all terms meta x comma meta y indeed
not0 parenthesis meta x < meta y end parenthesis infer meta y <= meta x
end lemma end math ]"

"[ math system Q proof of lemma fromNotLess reads
block any term meta x comma meta y end line
line ell a premise not0 parenthesis meta x < meta y end parenthesis end line
line ell big a premise meta x <= meta y end line
line ell c because 1rule repetition modus ponens ell a indeed ""; XXREP <
  not0 not0 parenthesis meta x <= meta y imply not0 meta x != meta y end parenthesis end line
line ell d because prop lemma remove double neg modus ponens ell c indeed
  meta x <= meta y imply not0 meta x != meta y end line
line ell e because 1rule mp modus ponens ell d modus ponens ell big a indeed
  not0 meta x != meta y end line
line ell f because prop lemma remove double neg modus ponens ell e indeed meta x = meta y end line
line ell g because lemma eqSymmetry modus ponens ell f indeed meta y = meta x end line
because lemma eqLeq modus ponens ell g indeed meta y <= meta x end line
line ell h end block

any term meta x comma meta y end line
line ell i because 1rule deduction modus ponens ell h indeed
  not0 meta x < meta y imply meta x <= meta y imply meta y <= meta x end line
line ell j premise not0 meta x < meta y end line
line ell k because 1rule mp modus ponens ell i modus ponens ell j indeed 
  meta x <= meta y imply meta y <= meta x end line
line ell l because prop lemma auto imply indeed meta y <= meta x imply meta y <= meta x end line
line ell m because axiom leqTotality indeed meta x <= meta y or0 meta y <= meta x end line
because prop lemma from disjuncts modus ponens ell m modus ponens ell k modus ponens ell l indeed 
  meta y <= meta x qed
end math ]"


"[  math in theory system Q lemma lemma toNotLess says
for all terms meta x comma meta y indeed
meta x <= meta y infer not0 meta y < meta x
end lemma end math ]"

"[ math system Q proof of lemma toNotLess reads
block any term meta x comma meta y end line
line ell a premise meta x <= meta y end line
line ell b premise meta y <= meta x end line
line ell c because lemma leqAntisymmetry modus ponens ell b modus ponens ell a indeed 
  meta y = meta x end line
because prop lemma add double neg modus ponens ell c indeed 
  not0 not0 meta y = meta x end line
line ell d end block

any term meta x comma meta y end line
line ell e because 1rule deduction modus ponens ell d indeed
  meta x <= meta y imply meta y <= meta x imply not0 not0 meta y = meta x end line
line ell f premise meta x <= meta y end line
line ell g because 1rule mp modus ponens ell e modus ponens ell f indeed
  meta y <= meta x imply not0 not0 meta y = meta x end line
line ell h because prop lemma add double neg modus ponens ell g indeed
  not0 not0 parenthesis meta y <= meta x imply not0 not0 meta y = meta x end parenthesis end line
line ell i because 1rule repetition modus ponens ell h indeed ""; XXREP and
  not0 parenthesis meta y <= meta x and0 not0 meta y = meta x end parenthesis end line
because 1rule repetition modus ponens ell i indeed ""; XXREP <
  not0 meta y < meta x qed
end math ]"




(*** LEQ ***)

"[  math in theory system Q lemma lemma leqLessEq says
for all terms meta x comma meta y indeed
meta x <= meta y infer meta x < meta y or0 meta x = meta y
end lemma end math ]"

"[ math system Q proof of lemma leqLessEq reads
block any term meta x comma meta y end line
line ell a premise meta x <= meta y end line
line ell b premise not0 meta x < meta y end line
line ell c because lemma fromNotLess modus ponens ell b indeed meta y <= meta x end line
because lemma leqAntisymmetry modus ponens ell a modus ponens ell c indeed 
  meta x = meta y end line
line ell d end block

any term meta x comma meta y end line
line ell big a because 1rule deduction modus ponens ell d indeed
  meta x <= meta y imply not0 meta x < meta y imply meta x = meta y end line
line ell big b premise meta x <= meta y end line
line ell big c because 1rule mp modus ponens ell big a modus ponens ell big b indeed
  not0 meta x < meta y imply meta x = meta y end line
because 1rule repetition modus ponens ell big c indeed meta x < meta y or0 meta x = meta y qed ""; XXREP or
end math ]"



"[  math in theory system Q lemma lemma lessLeq says
for all terms meta x comma meta y indeed 
meta x < meta y infer meta x <= meta y
end lemma end math ]"

"[ math system Q proof of lemma lessLeq reads
any term meta x comma meta y end line
line ell a premise meta x < meta y end line
line ell b because 1rule repetition modus ponens ell a indeed ""; XXREP <
  meta x <= meta y and0 not0 parenthesis meta x = meta y end parenthesis end line
because prop lemma first conjunct modus ponens ell b indeed meta x <= meta y qed
end math ]"

"[  math in theory system Q lemma lemma from leqGeq says
for all terms meta a comma meta x comma meta y indeed
meta x <= meta y imply meta a infer
meta y <= meta x imply meta a infer meta a
end lemma end math ]"

"[ math system Q proof of lemma from leqGeq reads
any term meta a comma meta x comma meta y end line
line ell a premise meta x <= meta y imply meta a end line
line ell b premise meta y <= meta x imply meta a end line
line ell c because axiom leqTotality indeed meta x <= meta y or0 meta y <= meta x end line
because prop lemma from disjuncts modus ponens ell c modus ponens ell a modus ponens ell b indeed meta a qed
end math ]"



"[  math in theory system Q lemma lemma subLessRight says
for all terms meta x comma meta y comma meta z indeed
meta x = meta y infer meta z < meta x infer meta z < meta y 
end lemma end math ]"

"[ math system Q proof of lemma subLessRight reads
any term meta x comma meta  y comma meta z end line
line ell a premise meta x = meta y end line
line ell b premise meta z < meta x end line
line ell c because 1rule repetition modus ponens ell b indeed ""; XXREP <
  meta z <= meta x and0 meta z != meta x end line
line ell d because prop lemma first conjunct modus ponens ell c indeed
  meta z <= meta x end line
line ell e because lemma subLeqRight modus ponens ell a modus ponens ell d indeed
  meta z <= meta y end line
line ell f because prop lemma second conjunct modus ponens ell c indeed
  meta z != meta x end line
line ell k because lemma subNeqRight modus ponens ell a modus ponens ell f indeed
  meta z != meta y end line
because prop lemma join conjuncts modus ponens ell e modus ponens ell k indeed 
  meta z < meta y qed
end math ]"

"[  math in theory system Q lemma lemma subLessLeft says
for all terms meta x comma meta y comma meta z indeed
meta x = meta y infer meta x < meta z infer meta y < meta z 
end lemma end math ]"

"[ math system Q proof of lemma subLessLeft reads
any term meta x comma meta  y comma meta z end line
line ell a premise meta x = meta y end line
line ell b premise meta x < meta z end line
line ell c because 1rule repetition modus ponens ell b indeed ""; XXREP <
  meta x <= meta z and0 meta x != meta z end line
line ell d because prop lemma first conjunct modus ponens ell c indeed
  meta x <= meta z end line
line ell e because lemma subLeqLeft modus ponens ell a modus ponens ell d indeed
  meta y <= meta z end line
line ell f because prop lemma second conjunct modus ponens ell c indeed
  meta x != meta z end line
line ell k because lemma subNeqLeft modus ponens ell a modus ponens ell f indeed
  meta y != meta z end line
because prop lemma join conjuncts modus ponens ell e modus ponens ell k indeed 
  meta y < meta z qed
end math ]"

"[  math in theory system Q lemma lemma leqLessTransitivity says
for all terms meta x comma meta y comma meta z indeed
meta x <= meta y infer meta y < meta z infer meta x < meta z 
end lemma end math ]"

"[ math system Q proof of lemma leqLessTransitivity reads
block any term meta x comma meta y comma meta z end line
line ell a premise meta x <= meta y end line
line ell b premise meta y <  meta z end line
line ell big a premise meta x = meta z end line
line ell c because prop lemma  first conjunct modus ponens ell b indeed meta y <= meta z end line
line ell d because prop lemma second conjunct modus ponens ell b indeed meta y != meta z end line
line ell f because lemma subLeqLeft modus ponens ell big a modus ponens ell a indeed meta z <= meta y end line
line ell g because lemma leqAntisymmetry modus ponens ell c modus ponens ell f indeed 
  meta y = meta z end line
because prop lemma from contradiction modus ponens ell g modus ponens ell d indeed
  meta x != meta z end line
line ell h end block

any term meta x comma meta y comma meta z end line
line ell i because 1rule deduction modus ponens ell h indeed
  meta x <= meta y imply meta y < meta z imply meta x = meta z imply meta x != meta z end line
line ell j premise meta x <= meta y end line
line ell k premise meta y < meta z end line
line ell l because prop lemma mp2 modus ponens ell i modus ponens ell j modus ponens ell k indeed
  meta x = meta z imply meta x != meta z end line
line ell m because prop lemma imply negation modus ponens ell l indeed
  meta x != meta z end line
line ell o because prop lemma  first conjunct modus ponens ell k indeed meta y <= meta z end line
line ell q because lemma leqTransitivity modus ponens ell j modus ponens ell o indeed 
  meta x <= meta z end line
because prop lemma join conjuncts modus ponens ell q modus ponens ell m indeed meta x < meta z qed
end math ]"

"[  math in theory system Q lemma lemma lessAddition says
for all terms meta x comma meta y comma meta z indeed
meta x < meta y infer meta x + meta z < meta y + meta z
end lemma end math ]"

"[ math system Q proof of lemma lessAddition reads
any term meta x comma meta y comma meta z end line
line ell a premise meta x < meta y end line
line ell b because lemma lessLeq modus ponens ell a indeed meta x <= meta y end line
line ell c because lemma leqAddition modus ponens ell b indeed meta x + meta z <= meta y + meta z end line
line ell d because lemma lessNeq modus ponens ell a indeed meta x != meta y end line
line ell e because lemma neqAddition modus ponens ell d indeed 
  meta x + meta z != meta y + meta z end line

because prop lemma join conjuncts modus ponens ell c modus ponens ell e indeed
 meta x + meta z < meta y + meta z qed
end math ]"

"[  math in theory system Q lemma lemma lessAdditionLeft says
for all terms meta x comma meta y comma meta z indeed
meta x < meta y infer meta z + meta x < meta z + meta y
end lemma end math ]"

"[ math system Q proof of lemma lessAdditionLeft reads
any term meta x comma meta y comma meta z end line
line ell a premise meta x < meta y end line
line ell b because lemma lessAddition modus ponens ell a indeed meta x + meta z < meta y + meta z end line

line ell c because axiom plusCommutativity indeed meta x + meta z = meta z + meta x end line
line ell d because lemma subLessLeft modus ponens ell c modus ponens ell b indeed 
  meta z + meta x < meta y + meta z end line

line ell e because axiom plusCommutativity indeed meta y + meta z = meta z + meta y end line
because lemma subLessRight modus ponens ell e modus ponens ell d indeed 
  meta z + meta x < meta z + meta y qed
end math ]"


"[  math in theory system Q lemma lemma leqPlus1 says
for all terms meta x comma meta y indeed
meta x <= meta y infer meta x < meta y + 1
end lemma end math ]"

"[ math system Q proof of lemma leqPlus1 reads
any term meta x comma meta y end line
line ell a premise meta x <= meta y end line

line ell b because lemma 0<1 indeed 0 < 1 end line
line ell c because lemma lessAdditionLeft modus ponens ell b indeed 
  meta y + 0 < meta y + 1 end line
line ell d because axiom plus0 indeed meta y + 0 = meta y end line
line ell e because lemma subLessLeft modus ponens ell d modus ponens ell c indeed
  meta y < meta y + 1 end line

because lemma leqLessTransitivity modus ponens ell a modus ponens ell e indeed
  meta x < meta y + 1 qed
end math ]"



"[  math in theory system Q lemma lemma leqAdditionLeft says
for all terms meta x comma meta y comma meta z indeed
meta x <= meta y infer meta z + meta x <= meta z + meta y
end lemma end math ]"

"[ math system Q proof of lemma leqAdditionLeft reads
any term meta x comma meta y comma meta z end line
line ell a premise meta x <= meta y end line
line ell b because lemma leqAddition modus ponens ell a indeed meta x + meta z <= meta y + meta z end line

line ell c because axiom plusCommutativity indeed meta x + meta z = meta z + meta x end line
line ell d because axiom plusCommutativity indeed meta y + meta z = meta z + meta y end line

line ell e because lemma subLeqLeft modus ponens ell c modus ponens ell b indeed 
  meta z + meta x <= meta y + meta z end line

because lemma subLeqRight modus ponens ell d modus ponens ell e indeed 
  meta z + meta x <= meta z + meta y qed
end math ]"




"[  math in theory system Q lemma lemma leqSubtraction says
for all terms meta x comma meta y comma meta z indeed
meta x + meta z <= meta y + meta z infer meta x <= meta y
end lemma end math ]"

"[ math system Q proof of lemma leqSubtraction reads
any term meta x comma meta y comma meta z end line
line ell a premise meta x + meta z <= meta y + meta z end line
line ell b because lemma leqAddition modus ponens ell a indeed
  meta x + meta z - meta z <= meta y + meta z - meta z end line

line ell c because lemma x=x+y-y indeed meta x = meta x + meta z - meta z end line
line ell d because lemma eqSymmetry modus ponens ell c indeed meta x + meta z - meta z = meta x end line

line ell e because lemma x=x+y-y indeed meta y = meta y + meta z - meta z end line
line ell f because lemma eqSymmetry modus ponens ell e indeed meta y + meta z - meta z = meta y end line

line ell g because lemma subLeqLeft modus ponens ell d modus ponens ell b indeed
  meta x <= meta y + meta z - meta z end line
because lemma subLeqRight modus ponens ell f modus ponens ell g indeed 
  meta x <= meta y qed
end math ]"

"[  math in theory system Q lemma lemma leqSubtractionLeft says
for all terms meta x comma meta y comma meta z indeed
meta z + meta x <= meta z + meta y infer meta x <= meta y
end lemma end math ]"

"[ math system Q proof of lemma leqSubtractionLeft reads
any term meta x comma meta y comma meta z end line
line ell a premise meta z + meta x <= meta z + meta y end line

line ell b because axiom plusCommutativity indeed meta z + meta x = meta x + meta z end line
line ell c because axiom plusCommutativity indeed meta z + meta y = meta y + meta z end line

line ell d because lemma subLeqLeft modus ponens ell b modus ponens ell a indeed
  meta x + meta z <= meta z + meta y end line
line ell e because lemma subLeqRight modus ponens ell c modus ponens ell d indeed
  meta x + meta z <= meta y + meta z end line

because lemma leqSubtraction modus ponens ell e indeed meta x <= meta y qed
end math ]"


"[  math in theory system Q lemma lemma negativeToLeft(Leq) says
for all terms meta x comma meta y comma meta z indeed
meta x <= meta y - meta z infer meta x + meta z <= meta y 
end lemma end math ]"

"[ math system Q proof of lemma negativeToLeft(Leq) reads
any term meta x comma meta y comma meta z end line
line ell a premise meta x <= meta y - meta z end line
line ell b because lemma leqAddition modus ponens ell a indeed meta x + meta z <= meta y - meta z + meta z end line
line ell c because lemma x=x+y-y indeed meta y = meta y + meta z - meta z end line
line ell e because lemma three2threeTerms indeed
  meta y + meta z - meta z = meta y - meta z + meta z end line
line ell f because lemma eqTransitivity modus ponens ell c modus ponens ell e indeed
  meta y = meta y - meta z + meta z end line
line ell g because lemma eqSymmetry modus ponens ell f indeed meta y - meta z + meta z = meta y end line
because lemma subLeqRight modus ponens ell g modus ponens ell b indeed meta x + meta z <= meta y qed
end math ]"

"[  math in theory system Q lemma lemma thirdGeq says
for all terms meta x comma meta y indeed
meta x <= ex3 and0 meta y <= ex3
end lemma end math ]"

"[ math system Q proof of lemma thirdGeq reads
block any term meta x comma meta y end line
line ell a premise meta x <= meta y end line
line ell b because axiom leqReflexivity indeed meta y <= meta y end line
line ell c because prop lemma join conjuncts modus ponens ell a modus ponens ell b indeed
  meta x <= meta y and0 meta y <= meta y end line
because 1rule exist intro at ex3 at meta y modus ponens ell c indeed meta x <= ex3 and0 meta y <= ex3 end line
line ell d end block

block any term meta x comma meta y end line
line ell e premise meta y <= meta x end line
line ell f because axiom leqReflexivity indeed meta x <= meta x end line
line ell g because prop lemma join conjuncts modus ponens ell f modus ponens ell e indeed
  meta x <= meta x and0 meta y <= meta x end line
because 1rule exist intro at ex3 at meta x modus ponens ell g indeed meta x <= ex3 and0 meta y <= ex3 end line
line ell h end block

any term meta x comma meta y end line
line ell i because 1rule deduction modus ponens ell d indeed 
  meta x <= meta y imply meta x <= ex3 and0 meta y <= ex3 end line
line ell j because 1rule deduction modus ponens ell h indeed 
  meta y <= meta x imply meta x <= ex3 and0 meta y <= ex3 end line
line ell k because axiom leqTotality indeed meta x <= meta y or0 meta y <= meta x end line

because prop lemma from disjuncts modus ponens ell k modus ponens ell i modus ponens ell j indeed
  meta x <= ex3 and0 meta y <= ex3 qed
end math ]"

"[  math in theory system Q lemma lemma leqNegated says
for all terms meta x comma meta y indeed
meta x <= meta y infer - meta y <= - meta x
end lemma end math ]"

"[ math system Q proof of lemma leqNegated reads
any term meta x comma meta y end line
line ell a premise meta x <= meta y end line
line ell b because lemma leqAddition modus ponens ell a indeed meta x - meta x <= meta y - meta x end line
line ell c because axiom negative indeed meta x - meta x = 0 end line
line ell d because lemma subLeqLeft modus ponens ell c modus ponens ell b indeed 
  0 <= meta y - meta x end line
line ell e because axiom plusCommutativity indeed
  meta y - meta x = - meta x + meta y end line
line ell f because lemma subLeqRight modus ponens ell e modus ponens ell d indeed
  0 <= - meta x + meta y end line
line ell g because lemma leqAddition modus ponens ell f indeed
  0 - meta y <= - meta x + meta y - meta y end line

line ell h because lemma plus0Left indeed 0 - meta y = - meta y end line
line ell i because lemma x=x+y-y indeed - meta x = - meta x + meta y - meta y end line
line ell j because lemma eqSymmetry modus ponens ell i indeed - meta x + meta y - meta y = - meta x end line

line ell k because lemma subLeqLeft modus ponens ell h modus ponens ell g indeed 
  - meta y <= - meta x + meta y - meta y end line
because lemma subLeqRight modus ponens ell j modus ponens ell k indeed 
  - meta y <= - meta x qed
end math ]"

"[  math in theory system Q lemma lemma addEquations(Leq) says
for all terms meta x comma meta y comma meta z comma meta u indeed
meta x <= meta y infer meta z <= meta u infer meta x + meta z <= meta y + meta u
end lemma end math ]"

"[ math system Q proof of lemma addEquations(Leq) reads
any term meta x comma meta y comma meta z comma meta u end line
line ell a premise meta x <= meta y end line
line ell b premise meta z <= meta u end line
line ell c because lemma leqAddition modus ponens ell a indeed
  meta x + meta z <= meta y + meta z end line
line ell d because lemma leqAdditionLeft modus ponens ell b indeed
  meta y + meta z <= meta y + meta u end line
because lemma leqTransitivity modus ponens ell c modus ponens ell d indeed 
  meta x + meta z <= meta y + meta u qed
end math ]"



(*** LESS ***)

"[  math in theory system Q lemma lemma leqNeqLess says
for all terms meta x comma meta y indeed
meta x <= meta y infer meta x != meta y infer meta x < meta y
end lemma end math ]"

"[ math system Q proof of lemma leqNeqLess reads
any term meta x comma meta y end line
line ell a premise meta x <= meta y end line
line ell b premise meta x != meta y end line
line ell c because prop lemma join conjuncts modus ponens ell a modus ponens ell b indeed
  meta x <= meta y and0 meta x != meta y end line
because 1rule repetition modus ponens ell c indeed meta x < meta y qed ""; XXREP <
end math ]"


"[ math in theory system Q lemma lemma lessMultiplication says
for all terms meta x comma meta y comma meta z indeed
0 < meta z infer meta x < meta y infer meta x * meta z < meta y * meta z 
end lemma end math ]"

"[ math system Q proof of lemma lessMultiplication reads
any term meta x comma meta y comma meta z end line
line ell a premise 0 < meta z end line
line ell b premise meta x < meta y end line

line ell c because lemma lessLeq modus ponens ell b indeed meta x <= meta y end line
line ell d because lemma lessLeq modus ponens ell a indeed 0 <= meta z end line
line ell e because lemma leqMultiplication modus ponens ell d modus ponens ell c indeed
  meta x * meta z <= meta y * meta z end line

line ell f because lemma lessNeq modus ponens ell b indeed meta x != meta y end line
line ell g because lemma lessNeq modus ponens ell a indeed 0 != meta z end line
line ell big a because lemma neqSymmetry modus ponens ell g indeed meta z != 0 end line

line ell h because lemma neqMultiplication modus ponens ell big a modus ponens ell f indeed 
  meta x * meta z != meta y * meta z end line

because lemma leqNeqLess modus ponens ell e modus ponens ell h indeed 
 meta x * meta z < meta y * meta z qed
end math ]"

"[  math in theory system Q lemma lemma lessMultiplicationLeft says
for all terms meta x comma meta y comma meta z indeed
0 < meta z infer meta x < meta y infer meta z * meta x < meta z * meta y
end lemma end math ]"

"[ math system Q proof of lemma lessMultiplicationLeft reads
any term meta x comma meta y comma meta z end line
line ell a premise 0 < meta z end line
line ell b premise meta x < meta y end line
line ell c because lemma lessMultiplication modus ponens ell a modus ponens ell b indeed
  meta x * meta z < meta y * meta z end line
line ell d because axiom timesCommutativity indeed meta x * meta z = meta z * meta x end line
line ell e because axiom timesCommutativity indeed meta y * meta z = meta z * meta y end line

line ell f because lemma subLessLeft modus ponens ell d modus ponens ell c indeed
  meta z * meta x < meta y * meta z end line
because lemma subLessRight modus ponens ell e modus ponens ell f indeed  
    meta z * meta x < meta z * meta y qed
end math ]"

"[  math in theory system Q lemma lemma lessDivision says
for all terms meta x comma meta y comma meta z indeed
0 <= meta z infer meta x * meta z < meta y * meta z infer meta x < meta y 
end lemma end math ]"



"[ math system Q proof of lemma lessDivision reads
any term meta x comma meta y comma meta z end line

line ell a premise 0 <= meta z end line
line ell b premise meta x * meta z < meta y * meta z end line
line ell c because lemma fromLess modus ponens ell b indeed 
  not0 meta y * meta z <= meta x * meta z end line


line ell d because axiom leqMultiplication indeed
   0 <= meta z imply meta y <= meta x imply meta y * meta z <= meta x * meta z end line
line ell e because 1rule mp modus ponens ell d modus ponens ell a indeed
   meta y <= meta x imply meta y * meta z <= meta x * meta z end line
line ell f because prop lemma contrapositive modus ponens ell e indeed
  not0 meta y * meta z <= meta x * meta z imply not0 meta y <= meta x end line

line ell g because 1rule mp modus ponens ell f modus ponens ell c indeed
  not0 meta y <= meta x end line

because lemma toLess modus ponens ell g indeed meta x < meta y qed
end math ]"





"[  math in theory system Q lemma lemma lessLeqTransitivity says
for all terms meta x comma meta y comma meta z indeed
meta x < meta y infer meta y <= meta z infer meta x < meta z 
end lemma end math ]"

"[ math system Q proof of lemma lessLeqTransitivity reads
block any term meta x comma meta y comma meta z end line
line ell a premise meta x < meta y end line
line ell b premise meta y <= meta z end line
line ell big a premise meta z = meta x end line
line ell c because prop lemma  first conjunct modus ponens ell a indeed meta x <= meta y end line
line ell d because prop lemma second conjunct modus ponens ell a indeed meta x != meta y end line
line ell f because lemma subLeqRight modus ponens ell big a modus ponens ell b indeed meta y <= meta x end line
line ell g because lemma leqAntisymmetry modus ponens ell c modus ponens ell f indeed 
  meta x = meta y end line
because prop lemma from contradiction modus ponens ell g modus ponens ell d indeed
  meta z != meta x end line
line ell h end block

any term meta x comma meta y comma meta z end line
line ell i because 1rule deduction modus ponens ell h indeed
  meta x < meta y imply meta y <= meta z imply meta z = meta x imply meta z !=  meta x end line
line ell j premise meta x < meta y end line
line ell k premise meta y <= meta z end line
line ell l because prop lemma mp2 modus ponens ell i modus ponens ell j modus ponens ell k indeed
  meta z = meta x imply meta z != meta x end line
line ell m because prop lemma imply negation modus ponens ell l indeed
  meta z != meta x end line
line ell n because lemma neqSymmetry modus ponens ell m indeed meta x != meta z end line
line ell o because prop lemma  first conjunct modus ponens ell j indeed meta x <= meta y end line
line ell q because lemma leqTransitivity modus ponens ell o modus ponens ell k indeed 
  meta x <= meta z end line
because prop lemma join conjuncts modus ponens ell q modus ponens ell n indeed meta x < meta z qed
end math ]"

"[  math in theory system Q lemma lemma lessTransitivity says
for all terms meta x comma meta y comma meta z indeed
meta x < meta y infer meta y < meta z infer meta x < meta z 
end lemma end math ]"

"[ math system Q proof of lemma lessTransitivity reads
any term meta x comma meta y comma meta z end line
line ell a premise meta x < meta y end line
line ell b premise meta y < meta z end line
line ell c because prop lemma first conjunct modus ponens ell b indeed meta y <= meta z end line

because lemma lessLeqTransitivity modus ponens ell a modus ponens ell c indeed meta x < meta z qed
end math ]"


"[  math in theory system Q lemma lemma addEquations(Less) says
for all terms meta x comma meta y comma meta z comma meta u indeed
meta x < meta y infer meta z < meta u infer meta x + meta z < meta y + meta u
end lemma end math ]"

"[ math system Q proof of lemma addEquations(Less) reads
any term meta x comma meta y comma meta z comma meta u end line
line ell a premise meta x < meta y end line
line ell b premise meta z < meta u end line
line ell c because lemma lessAddition modus ponens ell a indeed 
  meta x + meta z < meta y + meta z end line
line ell d because lemma lessAdditionLeft modus ponens ell b indeed
  meta y + meta z < meta y + meta u end line
because lemma lessTransitivity modus ponens ell c modus ponens ell d indeed  
  meta x + meta z < meta y + meta u qed
end math ]"

"[  math in theory system Q lemma lemma lessTotality says
for all terms meta x comma meta y indeed
meta x < meta y or0 meta x = meta y or0 meta y < meta x
end lemma end math ]"

"[ math system Q proof of lemma lessTotality reads
block any term meta x comma meta y end line
line ell a premise not0 meta x < meta y end line
line ell b premise meta x != meta y end line
line ell c because lemma fromNotLess modus ponens ell a indeed 
  meta y <= meta x end line
line ell d because lemma neqSymmetry modus ponens ell b indeed
  meta y != meta x end line
because lemma leqNeqLess modus ponens ell c  modus ponens ell d indeed meta y < meta x end line
line ell e end block

any term meta x comma meta y end line
line ell f because 1rule deduction modus ponens ell e indeed 
  not0 meta x < meta y imply meta x != meta y imply meta y < meta x end line
because 1rule repetition modus ponens ell f indeed ""; XXREP or
  meta x < meta y or0 meta x = meta y or0 meta y < meta x qed
  
end math ]"



"[  math in theory system Q lemma lemma negativeLessPositive says
for all terms meta x indeed
0 < meta x infer - meta x < meta x 
end lemma end math ]"

"[ math system Q proof of lemma negativeLessPositive reads
any term meta x end line
line ell a premise 0 < meta x end line
line ell b because prop lemma first conjunct modus ponens ell a indeed 0 <= meta x end line

line ell d because lemma leqAddition modus ponens ell b indeed  0 - meta x <= meta x - meta x end line
line ell e because lemma plus0Left indeed 0 - meta x = - meta x end line
line ell f because axiom negative indeed meta x - meta x = 0 end line
line ell g because lemma subLeqLeft modus ponens ell e modus ponens ell d indeed 
  - meta x <= meta x - meta x end line
line ell h because lemma subLeqRight modus ponens ell f modus ponens ell g indeed
  - meta x <= 0 end line

because lemma leqLessTransitivity modus ponens ell h modus ponens ell a indeed  
  - meta x < meta x qed end math ]"


"[  math in theory system Q lemma lemma lessNegated says
for all terms meta x comma meta y indeed
meta x < meta y infer - meta y < - meta x
end lemma end math ]"


"[ math system Q proof of lemma lessNegated reads
any term meta x comma meta y end line
line ell a premise meta x < meta y end line
line ell b because lemma lessLeq modus ponens ell a indeed 
  meta x <= meta y end line
line ell c because lemma leqNegated modus ponens ell b indeed
  - meta y <= - meta x end line

line ell d because lemma lessNeq modus ponens ell a indeed
  meta x != meta y end line
line ell e because lemma neqNegated modus ponens ell d indeed
  not0 - meta x = - meta y end line
line ell f because lemma neqSymmetry modus ponens ell e indeed
  not0 - meta y = - meta x end line

because lemma leqNeqLess modus ponens ell c modus ponens ell f indeed 
  - meta y < - meta x qed
end math ]"

"[  math in theory system Q lemma lemma -0=0 says - 0 =  0
end lemma end math ]"

"[ math system Q proof of lemma -0=0 reads
line ell a because axiom negative indeed 0 - 0 = 0 end line
line ell b because axiom plus0 indeed 0 + 0 = 0 end line
because lemma uniqueNegative modus ponens ell a modus ponens ell b indeed - 0 = 0 qed
end math ]"


"[  math in theory system Q lemma lemma positiveNegated says
for all terms meta x indeed
0 < meta x infer - meta x < 0
end lemma end math ]"

"[ math system Q proof of lemma positiveNegated reads
any term meta x end line
line ell a premise 0 < meta x end line
line ell b because lemma lessNegated modus ponens ell a indeed
  - meta x < - 0 end line
line ell c because lemma -0=0 indeed - 0 = 0 end line

because lemma subLessRight modus ponens ell c modus ponens ell b indeed - meta x < 0 qed
end math ]"

"[  math in theory system Q lemma lemma nonpositiveNegated says
for all terms meta x indeed
meta x <= 0 infer 0 <= - meta x
end lemma end math ]"

"[ math system Q proof of lemma nonpositiveNegated reads
any term meta x end line
line ell a premise meta x <= 0 end line
line ell b because lemma leqNegated modus ponens ell a indeed
  - 0 <= - meta x end line
line ell c because lemma -0=0 indeed - 0 = 0 end line
because lemma subLeqLeft modus ponens ell c modus ponens ell b indeed 0 <= - meta x qed
end math ]"

"[  math in theory system Q lemma lemma negativeNegated says
for all terms meta x indeed
meta x < 0 infer 0 < - meta x 
end lemma end math ]"

"[ math system Q proof of lemma negativeNegated reads
any term meta x end line
line ell a premise meta x < 0 end line
line ell b because lemma lessNegated modus ponens ell a indeed - 0 < - meta x end line
line ell c because lemma -0=0 indeed - 0 = 0 end line
because lemma subLessLeft modus ponens ell c modus ponens ell b indeed 0 < - meta x qed
end math ]"

"[  math in theory system Q lemma lemma nonnegativeNegated says
for all terms meta x indeed
0 <= meta x infer - meta x <= 0
end lemma end math ]"

"[ math system Q proof of lemma nonnegativeNegated reads
any term meta x end line
line ell a premise 0 <= meta x end line
line ell b because lemma leqNegated modus ponens ell a indeed - meta x <= - 0 end line
line ell c because lemma -0=0 indeed - 0 = 0 end line
because lemma subLeqRight modus ponens ell c modus ponens ell b indeed 
  - meta x <= 0 qed
end math ]"


"[  math in theory system Q lemma lemma 0<2 says
0 < 2 end lemma end math ]"

"[ math system Q proof of lemma 0<2 reads
line ell a because lemma 0<1 indeed 0 < 1 end line

"";line ell b because lemma lessLeq modus ponens ell a indeed 0 <= 1 end line

line ell c because lemma lessAddition modus ponens ell a indeed
  0 + 1 < 1 + 1 end line
line ell d because lemma plus0Left indeed 0 + 1 = 1 end line
line ell e because lemma subLessLeft modus ponens ell d modus ponens ell c indeed
  1 < 1 + 1 end line

because lemma lessTransitivity modus ponens ell a modus ponens ell e indeed  
  0 < 2 qed
end math ]"


"[  math in theory system Q lemma lemma 0<1/2 says
0 < 1/2 end lemma end math ]"

"[ math system Q proof of lemma 0<1/2 reads
line ell big a because lemma 0<2 indeed 0 < 2 end line
line ell big b because prop lemma first conjunct modus ponens ell big a indeed 0 <= 2 end line
line ell big c because prop lemma second conjunct modus ponens ell big a indeed 0 != 2 end line
line ell big d because lemma neqSymmetry modus ponens ell big c indeed 2 != 0 end line
line ell a because lemma 0<1 indeed 0 < 1 end line
line ell b because lemma x*0=0 indeed 2 * 0 = 0 end line
line ell d because lemma x*y=zBackwards  modus ponens ell b indeed 0 = 0 * 2 end line
line ell e because lemma subLessLeft modus ponens ell d modus ponens ell a indeed 
  0 * 2 < 1 end line
line ell f because lemma reciprocal modus ponens ell big d indeed 2 * 1/2 = 1 end line
line ell g because lemma x*y=zBackwards modus ponens ell f indeed 1 = 1/2 * 2 end line
line ell h because lemma subLessRight modus ponens ell g modus ponens ell e indeed
  0 * 2 < 1/2 * 2 end line
because lemma lessDivision modus ponens ell big b modus ponens ell h indeed 0 < 1/2 qed
end math ]"


"[  math in theory system Q lemma lemma positiveHalved says
for all terms meta x indeed
0 < meta x infer 0 < 1/2 * meta x 
end lemma end math ]"

"[ math system Q proof of lemma positiveHalved reads
any term meta x end line
line ell a premise 0 < meta x end line
line ell b because lemma 0<1/2 indeed 0 < 1/2 end line
line ell c because lemma lessMultiplicationLeft modus ponens ell b modus ponens ell a indeed
  1/2 * 0 < 1/2 * meta x end line
line ell d because lemma x*0=0 indeed 1/2 * 0 = 0 end line
because lemma subLessLeft modus ponens ell d modus ponens ell c indeed 0 < 1/2 * meta x qed
end math ]"


"[  math in theory system Q lemma lemma fromNot<< says
for all terms meta fx comma meta fy indeed
not0 R( meta fx ) << R( meta fy ) infer not0 meta fx <f meta fy
end lemma end math ]"

"[ math system Q proof of lemma fromNot<< reads
"";block any term meta fx comma meta fy end line
"";line ell a premise meta fx <f meta fy end line
"";because 1rule to<< modus ponens ell a indeed R( meta fx ) << R( meta fy ) end line
"";line ell big a end block

any term meta fx comma meta fy end line

line ell a because prop lemma auto imply indeed  
  meta fx <f meta fy imply R( meta fx ) << R( meta fy ) end line

line ell b premise not0 R( meta fx ) << R( meta fy ) end line

because prop lemma mt modus ponens ell a modus ponens ell b indeed not0 meta fx <f meta fy qed
end math ]"


(*** NUMERISK ***)

"[  math in theory system Q lemma prop lemma to negated and says
for all terms meta a comma meta b indeed
meta a imply not0 meta b infer not0 ( meta a and0 meta b ) 
end lemma end math ]"

"[ math system Q proof of prop lemma to negated and reads
any term meta a comma meta b end line
line ell a premise meta a imply not0 meta b end line
line ell b because prop lemma add double neg modus ponens ell a indeed
  not0 not0 ( meta a imply not0 meta b ) end line
because 1rule repetition modus ponens ell b indeed not0 ( meta a and0 meta b ) qed ""; XXREP and
end math ]"

"[ math in theory system Q lemma prop lemma to negated and(1) says
for all terms meta a comma meta b indeed
not0 meta a infer not0 ( meta a and0 meta b )
end lemma end math ]"

"[ math system Q proof of prop lemma to negated and(1) reads
block any term meta a comma meta b end line
line ell a premise not0 meta a end line
line ell b premise meta a end line 
because prop lemma from contradiction modus ponens ell b modus ponens ell a indeed
  not0 meta b end line
line ell big a end block

any term meta a comma meta b end line
line ell big b because 1rule deduction modus ponens ell big a indeed
  not0 meta a imply meta a imply not0 meta b end line
line ell a premise not0 meta a end line
line ell b because 1rule mp modus ponens ell big b modus ponens ell a indeed
  meta a imply not0 meta b end line
because prop lemma to negated and modus ponens ell b indeed not0 ( meta a and0 meta b ) qed
end math ]"

"[  math in theory system Q lemma lemma nonnegativeNumerical says
for all terms meta x indeed
0 <= meta x infer | meta x | = meta x
end lemma end math ]"

"[ math system Q proof of lemma nonnegativeNumerical reads
any term meta x end line
line ell a premise 0 <= meta x end line
line ell b because axiom numerical indeed
  ( 0 <= meta x      and0 | meta x | = meta x   ) or0 
  ( not0 0 <= meta x and0 | meta x | = - meta x ) end line
line ell c because prop lemma add double neg modus ponens ell a indeed not0 not0 0 <= meta x end line
line ell d because prop lemma to negated and(1) modus ponens ell c indeed
  not0 ( not0 0 <= meta x and0 | meta x | = - meta x ) end line
line ell e because prop lemma negate second disjunct modus ponens ell b modus ponens ell d indeed
   0 <= meta x      and0 | meta x | = meta x  end line
because prop lemma second conjunct modus ponens ell e indeed | meta x | = meta x qed
end math ]"

"[  math in theory system Q lemma lemma positiveNumerical says
for all terms meta x indeed
0 < meta x infer | meta x | = meta x
end lemma end math ]"

"[  math system Q proof of lemma positiveNumerical reads
any term meta x end line
line ell a premise 0 < meta x end line
line ell b because lemma lessLeq modus ponens ell a indeed 0 <= meta x end line
because lemma nonnegativeNumerical modus ponens ell b indeed | meta x | = meta x qed
end math ]"

"[  math in theory system Q lemma lemma negativeNumerical says
for all terms meta x indeed
meta x < 0 infer | meta x | = - meta x 
end lemma end math ]"

"[ math system Q proof of lemma negativeNumerical reads
any term meta x end line
line ell a premise meta x < 0 end line
line ell b because axiom numerical indeed
  ( 0 <= meta x      and0 | meta x | = meta x   ) or0 
  ( not0 0 <= meta x and0 | meta x | = - meta x ) end line
line ell c because lemma fromLess modus ponens ell a indeed not0 0 <= meta x end line
line ell d because prop lemma to negated and(1) modus ponens ell c indeed
  not0 ( 0 <= meta x and0 | meta x | = meta x ) end line
line ell e because prop lemma negate first disjunct modus ponens ell b modus ponens ell d indeed
  not0 0 <= meta x and0 | meta x | = - meta x end line
because prop lemma second conjunct modus ponens ell e indeed | meta x | = - meta x qed
end math ]"

"[  math in theory system Q lemma lemma nonpositiveNumerical says
for all terms meta x indeed
meta x <= 0 infer | meta x | = - meta x
end lemma end math ]"

"[ math system Q proof of lemma nonpositiveNumerical reads

block any term meta x end line
line ell a premise meta x < 0 end line
because lemma negativeNumerical modus ponens ell a indeed | meta x | = - meta x end line
line ell b end block

block any term meta x end line
line ell c premise meta x = 0 end line

line ell d because lemma eqSymmetry modus ponens ell c indeed 0 = meta x end line
line ell e because lemma eqLeq modus ponens ell d indeed 0 <= meta x end line
line ell f because lemma nonnegativeNumerical modus ponens ell e indeed | meta x | = meta x end line

line ell g because lemma -0=0 indeed - 0 = 0 end line
line ell h because lemma eqSymmetry modus ponens ell g indeed 0 = - 0 end line

line ell i because lemma eqNegated modus ponens ell d indeed - 0 = - meta x end line

because lemma eqTransitivity5 modus ponens ell f modus ponens ell c modus ponens ell h modus ponens ell i indeed | meta x | = - meta x end line
line ell j end block

any term meta x end line
line ell big a because 1rule deduction modus ponens ell b indeed
  meta x < 0 imply | meta x | = - meta x end line
line ell big b because 1rule deduction modus ponens ell j indeed
  meta x = 0 imply | meta x | = - meta x end line
line ell big c premise meta x <= 0 end line
line ell big d because lemma leqLessEq modus ponens ell big c indeed meta x < 0 or0 meta x = 0 end line

because prop lemma from disjuncts modus ponens ell big d modus ponens ell big a modus ponens ell big b 
  indeed | meta x | = - meta x qed
end math ]"


"[  math in theory system Q lemma lemma |0|=0 says
| 0 | = 0 end lemma end math ]"

"[ math system Q proof of lemma |0|=0 reads
line ell a because axiom leqReflexivity indeed 0 <= 0 end line
because lemma nonnegativeNumerical modus ponens ell a indeed
  | 0 | = 0 qed
end math ]"



"[  math in theory system Q lemma lemma 0<=|x| says
for all terms meta x indeed
0 <= | meta x |
end lemma end math ]"

"[ math system Q proof of lemma 0<=|x| reads
block any term meta x end line
line ell a premise 0 <= meta x end line
line ell b because lemma nonnegativeNumerical modus ponens ell a indeed | meta x | = meta x end line
line ell c because lemma eqSymmetry modus ponens ell b indeed meta x = | meta x | end line
because lemma subLeqRight modus ponens ell c modus ponens ell a indeed 0 <= | meta x | end line
line ell d end block 

block any term meta x end line
line ell e premise not0 0 <= meta x end line
line ell f because lemma toLess modus ponens ell e indeed meta x < 0 end line
line ell g because lemma negativeNumerical modus ponens ell f indeed | meta x | = - meta x end line
line ell h because lemma eqSymmetry modus ponens ell g indeed - meta x = | meta x | end line
line ell i because lemma negativeNegated modus ponens ell f indeed 0 < - meta x end line
line ell j because lemma lessLeq modus ponens ell i indeed 0 <= - meta x end line
because lemma subLeqRight modus ponens ell h modus ponens ell j indeed 0 <= | meta x | end line
line ell k end block

any term meta x end line
line ell l because 1rule deduction modus ponens ell d indeed 0 <= meta x imply 0 <= | meta x | end line
line ell m because 1rule deduction modus ponens ell k indeed not0 0 <= meta x imply 0 <= | meta x | end line

because prop lemma from negations modus ponens ell l modus ponens ell m indeed 0 <= | meta x | qed
end math ]"


"[  math in theory system Q lemma lemma sameNumerical says
for all terms meta x comma meta y indeed
meta x = meta y infer | meta x | = | meta y |
end lemma end math ]"

"[ math system Q proof of lemma sameNumerical reads
block any term meta x comma meta y end line
line ell big a premise 0 <= meta x end line
line ell big b premise meta x = meta y end line
line ell big c because lemma nonnegativeNumerical modus ponens ell big a indeed
  | meta x | = meta x end line

line ell big d because lemma subLeqRight modus ponens ell big b modus ponens ell big a indeed  
  0 <= meta y end line
line ell big e because lemma nonnegativeNumerical modus ponens ell big d indeed
  | meta y | = meta y end line
line ell big f because lemma eqSymmetry modus ponens ell big e indeed meta y = | meta y | end line

because lemma eqTransitivity4 modus ponens ell big c modus ponens ell big b modus ponens ell 
  big f indeed | meta x | = | meta y | end line
line ell big g end block
	

block any term meta x comma meta y end line
line ell d premise not0 0 <= meta x end line
line ell e premise meta x = meta y end line

line ell f because lemma toLess modus ponens ell d indeed meta x < 0 end line
line ell g because lemma negativeNumerical modus ponens ell f indeed
  | meta x | = - meta x end line

line ell h because lemma subLessLeft modus ponens ell e modus ponens ell f indeed meta y < 0 end line
line ell i because lemma negativeNumerical modus ponens ell h indeed
  | meta y | = - meta y end line
line ell j because lemma eqSymmetry modus ponens ell i indeed - meta y = | meta y | end line

line ell k because lemma eqNegated modus ponens ell e indeed - meta x = - meta y end line
because lemma eqTransitivity4 modus ponens ell g modus ponens ell k modus ponens ell j indeed | meta x | = | meta y | end line
line ell l end block

any term meta x comma meta y end line
line ell m premise meta x = meta y end line
line ell n because 1rule deduction modus ponens ell big g indeed
  0 <= meta x imply meta x = meta y imply | meta x | = | meta y | end line
line ell o because 1rule deduction modus ponens ell l indeed
  not0 0 <= meta x imply meta x = meta y imply | meta x | = | meta y | end line
line ell p because prop lemma from negations modus ponens ell n modus ponens ell o indeed
  meta x = meta y imply | meta x | = | meta y | end line

because 1rule mp modus ponens ell p modus ponens ell m indeed | meta x | = | meta y | qed
end math ]"

"[  math in theory system Q lemma lemma signNumerical(+) says
for all terms meta x indeed
0 < meta x infer | meta x | = | - meta x |
end lemma end math ]"

"[ math system Q proof of lemma signNumerical(+) reads
any term meta x end line
line ell a premise 0 < meta x end line
line ell b because lemma positiveNumerical modus ponens ell a indeed 
  | meta x | = meta x end line
line ell c because lemma positiveNegated modus ponens ell a indeed - meta x < 0 end line
line ell d because lemma negativeNumerical modus ponens ell c indeed
  | - meta x | = - - meta x end line
line ell e because lemma doubleMinus indeed - - meta x = meta x end line
line ell f because lemma eqTransitivity modus ponens ell d modus ponens ell e indeed
  | - meta x | = meta x end line
line ell g because lemma eqSymmetry modus ponens ell f indeed 
  meta x = | - meta x | end line

because lemma eqTransitivity modus ponens ell b modus ponens ell g indeed 
  | meta x | = | - meta x | qed
end math ]"


"[  math in theory system Q lemma lemma signNumerical says
for all terms meta x indeed
| meta x | = | - meta x |
end lemma end math ]"

"[ math system Q proof of lemma signNumerical reads

block any term meta x end line
line ell a premise meta x < 0 end line
line ell b because lemma negativeNegated modus ponens ell a indeed 0 < - meta x end line
line ell c because lemma signNumerical(+) modus ponens ell b indeed
  | - meta x | = | - - meta x | end line
line ell d because lemma doubleMinus indeed - - meta x = meta x end line
line ell e because lemma sameNumerical modus ponens ell d indeed | - - meta x | = | meta x | end line
line ell f because lemma eqTransitivity modus ponens ell c modus ponens ell e indeed
  | - meta x | = | meta x | end line
because lemma eqSymmetry modus ponens ell f indeed | meta x | = | - meta x | end line
line ell big a end block


block any term meta x end line
line ell a premise meta x = 0 end line

line ell b because lemma eqNegated modus ponens ell a indeed - meta x = - 0 end line
line ell c because lemma -0=0 indeed - 0 = 0 end line
line ell d because lemma eqSymmetry modus ponens ell a indeed 0 = meta x end line

line ell e because lemma eqTransitivity4 modus ponens ell b modus ponens ell c 
  modus ponens ell d indeed - meta x = meta x end line

line ell f because lemma eqSymmetry modus ponens ell e indeed  meta x = - meta x end line
because lemma sameNumerical modus ponens ell f indeed | meta x | = | - meta x | end line
line ell big b end block


block any term meta x end line
line ell a premise  0 < meta x end line
because lemma signNumerical(+) modus ponens ell a indeed 
  | meta x | = | - meta x | end line
line ell big c end block

any term meta x end line
line ell big d because 1rule deduction modus ponens ell big a indeed
  meta x < 0 imply | meta x | = | - meta x | end line 
line ell big e because 1rule deduction modus ponens ell big b indeed
  meta x = 0 imply | meta x | = | - meta x | end line
line ell big f because 1rule deduction modus ponens ell big c indeed
  0 < meta x imply | meta x | = | - meta x | end line

line ell big g because lemma lessTotality indeed
  meta x < 0 or0 meta x = 0 or0 0 < meta x end line

because prop lemma from three disjuncts modus ponens ell big g 
  modus ponens ell big d modus ponens ell big e modus ponens ell big f indeed 
  | meta x | = | - meta x | qed
end math ]"




"[  math in theory system Q lemma lemma -x-y=-(x+y) says
for all terms meta x comma meta y indeed
- meta x - meta y = - parenthesis meta x + meta y end parenthesis 
end lemma end math ]"

"[ math system Q proof of lemma -x-y=-(x+y) reads
any term meta x comma meta y end line

line ell a because lemma times(-1)Left indeed (-1) * meta x = - meta x end line
line ell b because lemma times(-1)Left indeed (-1) * meta y = - meta y end line
line ell c because lemma addEquations modus ponens ell a modus ponens ell b indeed
  (-1) * meta x + (-1) * meta y = - meta x - meta y end line
line ell d because lemma eqSymmetry modus ponens ell c indeed
 - meta x - meta y = (-1) * meta x + (-1) * meta y end line

line ell e because lemma distributionOut indeed
  (-1) * meta x + (-1) * meta y = (-1) * parenthesis meta x + meta y end parenthesis end line

line ell f because lemma times(-1)Left indeed
  (-1) * parenthesis meta x + meta y end parenthesis = 
  - parenthesis meta x + meta y end parenthesis end line

because lemma eqTransitivity4 modus ponens ell d modus ponens ell e modus ponens ell f 
  indeed - meta x - meta y = - parenthesis meta x + meta y end parenthesis qed
end math ]"


"[  math in theory system Q lemma lemma minusNegated says
for all terms meta x comma meta y indeed
- parenthesis meta x - meta y end parenthesis = meta y - meta x
end lemma end math ]"

"[ math system Q proof of lemma minusNegated reads
any term meta x comma meta y end line
line ell a because lemma doubleMinus indeed - - meta y = meta y end line
line ell b because lemma eqAddition modus ponens ell a indeed - - meta y - meta x = meta y - meta x end line
line ell c because lemma eqSymmetry modus ponens ell b indeed 
  meta y - meta x = - - meta y - meta x end line

line ell d because lemma -x-y=-(x+y) indeed
  - - meta y - meta x = - parenthesis - meta y + meta x end parenthesis end line

line ell e because axiom plusCommutativity indeed - meta y + meta x = meta x - meta y end line
line ell f because lemma eqNegated modus ponens ell e indeed
  - parenthesis - meta y + meta x end parenthesis = - parenthesis meta x - meta y end parenthesis end line

line ell g because lemma eqTransitivity4 modus ponens ell c modus ponens ell d modus ponens ell f indeed
  meta y - meta x = - parenthesis meta x - meta y end parenthesis end line

because lemma eqSymmetry modus ponens ell g indeed 
  - parenthesis meta x - meta y end parenthesis = meta y - meta x qed
end math ]"


"[  math in theory system Q lemma lemma numericalDifference says
for all terms meta x comma meta y indeed
| meta x - meta y | = | meta y - meta x |
end lemma end math ]"

"[ math system Q proof of lemma numericalDifference reads
any term meta x comma meta y end line
line ell a because lemma signNumerical indeed 
  | meta x - meta y | = | - parenthesis meta x - meta y end parenthesis | end line

line ell b because lemma minusNegated indeed
  - parenthesis meta x - meta y end parenthesis = meta y - meta x end line
line ell c because lemma sameNumerical modus ponens ell b indeed
  | - parenthesis meta x - meta y end parenthesis | = | meta y - meta x | end line
  
because lemma eqTransitivity modus ponens ell a modus ponens ell c indeed 
| meta x - meta y | = | meta y - meta x | qed
end math ]"

"[  math in theory system Q lemma lemma splitNumericalSumHelper says
for all terms meta x comma meta y indeed
| - meta x - meta y | <= | - meta x | + | - meta y | infer 


|   meta x + meta y | <= |   meta x | + |   meta y |
end lemma end math ]"

"[ math system Q proof of lemma splitNumericalSumHelper reads
any term meta x comma meta y end line
line ell a premise | - meta x - meta y | <= | - meta x | + | - meta y | end line

line ell b because lemma signNumerical indeed | meta x | = | - meta x | end line
line ell c because lemma signNumerical indeed | meta y | = | - meta y | end line
line ell d because lemma addEquations modus ponens ell b modus ponens ell c indeed
  | meta x | + | meta y | = | - meta x | + | - meta y | end line
line ell e because lemma eqSymmetry modus ponens ell d indeed
  | - meta x | + | - meta y | = | meta x | + | meta y | end line

line ell f because lemma -x-y=-(x+y) indeed
  - meta x - meta y = - parenthesis meta x + meta y end parenthesis end line
line ell g because lemma sameNumerical modus ponens ell f indeed
  | - meta x - meta y | = | - parenthesis meta x + meta y end parenthesis | end line

line ell h because lemma signNumerical indeed 
  | meta x + meta y | = | - parenthesis meta x + meta y end parenthesis | end line
line ell i because lemma eqSymmetry modus ponens ell h indeed
  | - parenthesis meta x + meta y end parenthesis | = | meta x + meta y | end line

line ell j because lemma eqTransitivity modus ponens ell g modus ponens ell i indeed
  | - meta x - meta y | = | meta x + meta y | end line

line ell k because lemma subLeqRight modus ponens ell e modus ponens ell a indeed
  | - meta x - meta y | <= | meta x | + | meta y | end line
because lemma subLeqLeft modus ponens ell j modus ponens ell k indeed 
  |   meta x + meta y | <= | meta x | + | meta y | qed
end math ]"


"[ math in theory system Q lemma lemma splitNumericalSum(++) says
for all terms meta x comma meta y indeed
0 <= meta x infer 0 <= meta y infer | meta x + meta y | <= | meta x | + | meta y |
end lemma end math ]"

"[ math system Q proof of lemma splitNumericalSum(++) reads
any term meta x comma meta y end line
line ell a premise 0 <= meta x end line
line ell b premise 0 <= meta y end line

line ell c because lemma addEquations(Leq) modus ponens ell a modus ponens ell b indeed
  0 + 0 <= meta x + meta y end line
line ell d because axiom plus0 indeed 0 + 0 = 0 end line
line ell e because lemma subLeqLeft modus ponens ell d modus ponens ell c indeed
  0 <= meta x + meta y end line
line ell f because lemma nonnegativeNumerical modus ponens ell e indeed
  | meta x + meta y | = meta x + meta y end line

line ell big a because lemma nonnegativeNumerical modus ponens ell a indeed
  | meta x | = meta x end line 
line ell big b because lemma nonnegativeNumerical modus ponens ell b indeed
  | meta y | = meta y end line
line ell big c because lemma addEquations modus ponens ell big a modus ponens ell big b indeed
  | meta x | + | meta y | = meta x + meta y end line
line ell big d because lemma eqSymmetry modus ponens ell big c indeed
  meta x + meta y = | meta x | + | meta y | end line

line ell big e because lemma eqTransitivity modus ponens ell f modus ponens ell big d indeed 
  | meta x + meta y | = | meta x | + | meta y | end line
because lemma eqLeq modus ponens ell big e indeed | meta x + meta y | <= | meta x | + | meta y | qed
end math ]"

"[  math in theory system Q lemma lemma splitNumericalSum(--) says
for all terms meta x comma meta y indeed
meta x <= 0 infer meta y <= 0 infer | meta x + meta y | <= | meta x | + | meta y |
end lemma end math ]"

"[ math system Q proof of lemma splitNumericalSum(--) reads
any term meta x comma meta y end line
line ell a premise meta x <= 0 end line
line ell b premise meta y <= 0 end line
line ell c because lemma nonpositiveNegated modus ponens ell a indeed 0 <= - meta x end line
line ell d because lemma nonpositiveNegated modus ponens ell b indeed 0 <= - meta y end line
line ell e because lemma splitNumericalSum(++) modus ponens ell c modus ponens ell d indeed
  | - meta x - meta y | <= | - meta x | + | - meta y | end line
because lemma splitNumericalSumHelper modus ponens ell e indeed
  | meta x + meta y | <= | meta x | + | meta y | qed
end math ]"

"[ math in theory system Q lemma lemma splitNumericalSum(+-, smallNegative) says
for all terms meta x comma meta y indeed
0 <= meta x infer meta y <= 0 infer | meta y | <= | meta x | infer | meta x + meta y | <= | meta x | 
end lemma end math ]"

"[ math system Q proof of lemma splitNumericalSum(+-, smallNegative) reads 
any term meta x comma meta y end line
line ell a premise 0 <= meta x end line
line ell b premise meta y <= 0 end line
line ell c premise | meta y | <= | meta x | end line

line ell big a because lemma leqAdditionLeft modus ponens ell b indeed
  meta x + meta y <= meta x + 0 end line
line ell big b because axiom plus0 indeed meta x + 0 = meta x end line
line ell big c because lemma subLeqRight modus ponens ell big b modus ponens ell big a indeed
  meta x + meta y <= meta x end line

line ell d because lemma positiveToRight(Leq)(1 term) modus ponens ell c indeed 
  0 <= | meta x | - | meta y | end line

line ell f because lemma nonpositiveNumerical modus ponens ell b indeed
  | meta y | = - meta y end line
line ell g because lemma eqNegated modus ponens ell f indeed
  - | meta y | = - - meta y end line
line ell h because lemma doubleMinus indeed - - meta y = meta y end line
line ell i because lemma eqTransitivity modus ponens ell g modus ponens ell h indeed
  - | meta y | = meta y end line

line ell e because lemma nonnegativeNumerical modus ponens ell a indeed
  | meta x | = meta x end line
line ell j because lemma addEquations modus ponens ell e modus ponens ell i indeed
  | meta x | - | meta y | = meta x + meta y end line
line ell k because lemma subLeqRight modus ponens ell j modus ponens ell d indeed 
  0 <= meta x + meta y end line
line ell l because lemma nonnegativeNumerical modus ponens ell k indeed
  | meta x + meta y | = meta x + meta y end line
line ell m because lemma eqSymmetry modus ponens ell l indeed
  meta x + meta y = | meta x + meta y | end line

line ell big q because lemma eqSymmetry modus ponens ell e indeed
  meta x = | meta x | end line

line ell n because lemma subLeqLeft modus ponens ell m modus ponens ell big c indeed 
  | meta x + meta y | <= meta x end line
because lemma subLeqRight modus ponens ell big q modus ponens ell n indeed  
  | meta x + meta y | <= | meta x | qed
end math ]"

"[  math in theory system Q lemma lemma splitNumericalSum(+-, bigNegative) says
for all terms meta x comma meta y indeed
0 <= meta x infer meta y <= 0 infer | meta x | < | meta y | infer | meta x + meta y | <= | meta y | 
end lemma end math ]"

"[ math system Q proof of lemma splitNumericalSum(+-, bigNegative) reads
any term meta x comma meta y end line
line ell a premise 0 <= meta x end line
line ell b premise meta y <= 0 end line
line ell c premise | meta x | < | meta y | end line

line ell big a because lemma nonnegativeNegated modus ponens ell a indeed - meta x <= 0 end line
line ell big b because lemma nonpositiveNegated modus ponens ell b indeed 0 <= - meta y end line

line ell big c because lemma signNumerical indeed | meta x | = | - meta x | end line
line ell big d because lemma subLessLeft modus ponens ell big c modus ponens ell c indeed
  | - meta x | < | meta y | end line
line ell big e because lemma signNumerical indeed | meta y | = | - meta y | end line
line ell big f because lemma subLessRight modus ponens ell big e modus ponens ell big d indeed
  | - meta x | < | - meta y | end line
line ell big g because lemma lessLeq modus ponens ell big f indeed
  | - meta x | <= | - meta y | end line

line ell big h because lemma splitNumericalSum(+-, smallNegative) modus ponens ell big b modus ponens ell big a modus ponens ell big g indeed 
| - meta y - meta x | <= | - meta y | end line

line ell d because lemma signNumerical indeed 
  | meta x + meta y | = | - parenthesis meta x + meta y end parenthesis | end line

line ell e because lemma -x-y=-(x+y) indeed
  - meta x - meta y = - parenthesis meta x + meta y end parenthesis end line
line ell f because axiom plusCommutativity indeed - meta x - meta y = - meta y - meta x end line
line ell g because lemma equality modus ponens ell e modus ponens ell f indeed
  - parenthesis meta x + meta y end parenthesis = - meta y - meta x end line
line ell h because lemma sameNumerical modus ponens ell g indeed
    | - parenthesis meta x + meta y end parenthesis | = | - meta y - meta x | end line
line ell i because lemma eqTransitivity modus ponens ell d modus ponens ell h indeed
  | meta x + meta y | = | - meta y - meta x | end line
line ell j because lemma eqSymmetry modus ponens ell i indeed
  | - meta y - meta x | = | meta x + meta y | end line

line ell k because lemma eqSymmetry modus ponens ell big e indeed | - meta y | = | meta y | end line

line ell l because lemma subLeqLeft modus ponens ell j modus ponens ell big h indeed
  | meta x + meta y | <= | - meta y | end line
because lemma subLeqRight modus ponens ell k modus ponens ell l indeed | meta x + meta y | <= | meta y | qed
end math ]"

"[  math in theory system Q lemma lemma splitNumericalSum(+-) says
for all terms meta x comma meta y indeed
0 <= meta x infer meta y <= 0 infer | meta x + meta y | <= | meta x | + | meta y |
end lemma end math ]"

"[ math system Q proof of lemma splitNumericalSum(+-) reads
block any term meta x comma meta y end line
line ell c premise | meta y | <= | meta x | end line
line ell a premise 0 <= meta x end line
line ell b premise meta y <= 0 end line
line ell d because lemma splitNumericalSum(+-, smallNegative) modus ponens ell a modus ponens ell b
  modus ponens ell c indeed | meta x + meta y | <= | meta x | end line

line ell e because lemma 0<=|x| indeed 0 <= | meta y | end line
line ell f because lemma leqAdditionLeft modus ponens ell e indeed 
  | meta x | + 0 <= | meta x | + | meta y | end line
line ell g because axiom plus0 indeed | meta x | + 0 = | meta x | end line
line ell h because lemma subLeqLeft modus ponens ell g modus ponens ell f indeed 
  | meta x | <= | meta x | + | meta y | end line
because lemma leqTransitivity modus ponens ell d modus ponens ell h indeed
  | meta x + meta y | <= | meta x | + | meta y | end line
line ell i end block

block any term meta x comma meta y end line
line ell big c premise not0 | meta y | <= | meta x | end line
line ell big a premise 0 <= meta x end line
line ell big z premise meta y <= 0 end line
line ell big d because lemma toLess modus ponens ell big c indeed | meta x | < | meta y | end line
line ell big e because lemma splitNumericalSum(+-, bigNegative) modus ponens ell big a modus ponens ell big z modus ponens ell big d indeed | meta x + meta y | <= | meta y | end line

line ell big f because lemma 0<=|x| indeed 0 <= | meta x | end line
line ell big g because lemma leqAddition modus ponens ell big f indeed 
  0 + | meta y | <= | meta x | + | meta y | end line
line ell big h because lemma plus0Left indeed 0 + | meta y | = | meta y | end line
line ell big i because lemma subLeqLeft modus ponens ell big h modus ponens ell big g indeed 
  | meta y | <= | meta x | + | meta y | end line
because lemma leqTransitivity modus ponens ell big e modus ponens ell big i indeed
  | meta x + meta y | <= | meta x | + | meta y | end line
line ell big j end block


any term meta x comma meta y end line
line ell m because 1rule deduction modus ponens ell i indeed
  | meta y | <= | meta x | imply 0 <= meta x imply meta y <= 0 imply
  | meta x + meta y | <= | meta x | + | meta y | end line
line ell n because 1rule deduction modus ponens ell big j indeed
  not0 | meta y | <= | meta x | imply 0 <= meta x imply meta y <= 0 imply 
  | meta x + meta y | <= | meta x | + | meta y | end line
line ell o premise 0 <= meta x end line
line ell p premise meta y <= 0 end line
line ell q because prop lemma from negations modus ponens ell m modus ponens ell n indeed
  0 <= meta x imply meta y <= 0 imply | meta x + meta y | <= | meta x | + | meta y | end line
because prop lemma mp2 modus ponens ell q modus ponens ell o modus ponens ell p indeed 

  | meta x + meta y | <= | meta x | + | meta y | qed
end math ]"

"[  math in theory system Q lemma lemma splitNumericalSum(-+) says
for all terms meta x comma meta y indeed
meta x <= 0 infer 0 <= meta y infer | meta x + meta y | <= | meta x | + | meta y |
end lemma end math ]"

"[ math system Q proof of lemma splitNumericalSum(-+) reads
any term meta x comma meta y end line
line ell a premise meta x <= 0 end line
line ell b premise 0 <= meta y end line
line ell c because lemma nonpositiveNegated modus ponens ell a indeed 0 <= - meta x end line
line ell d because lemma nonnegativeNegated modus ponens ell b indeed - meta y <= 0 end line
line ell e because lemma splitNumericalSum(+-) modus ponens ell c modus ponens ell d indeed
  | - meta x - meta y | <= | - meta x | + | - meta y | end line
because lemma splitNumericalSumHelper modus ponens ell e indeed 
   | meta x + meta y | <= | meta x | + | meta y | qed
end math ]"

"[  math in theory system Q lemma lemma splitNumericalSum says
for all terms meta x comma meta y indeed
| meta x + meta y | <= | meta x | + | meta y |
end lemma end math ]"

"[ math system Q proof of lemma splitNumericalSum reads
block any term meta x comma meta y end line
line ell a premise 0 <= meta x end line
line ell b premise 0 <= meta y end line
because lemma splitNumericalSum(++) modus ponens ell a modus ponens ell b indeed
  | meta x + meta y | <= | meta x | + | meta y | end line
line ell c end block 

block any term meta x comma meta y end line
line ell d premise  0 <= meta x end line
line ell e premise  meta y <= 0 end line
because lemma splitNumericalSum(+-) modus ponens ell d modus ponens ell e indeed
  | meta x + meta y | <= | meta x | + | meta y | end line
line ell f end block 

block any term meta x comma meta y end line
line ell g premise meta x <= 0 end line
line ell h premise 0 <= meta y end line
because lemma splitNumericalSum(-+) modus ponens ell g modus ponens ell h indeed
  | meta x + meta y | <= | meta x | + | meta y | end line
line ell i end block 

block any term meta x comma meta y end line
line ell j premise meta x <= 0 end line
line ell k premise meta y <= 0 end line
because lemma splitNumericalSum(--) modus ponens ell j modus ponens ell k indeed
  | meta x + meta y | <= | meta x | + | meta y | end line
line ell l end block

any term meta x comma meta y end line
line ell big a because 1rule deduction modus ponens ell c indeed
  0 <= meta x imply 0 <= meta y imply | meta x + meta y | <= | meta x | + | meta y | end line
line ell big b because 1rule deduction modus ponens ell f indeed
  0 <= meta x imply meta y <= 0 imply | meta x + meta y | <= | meta x | + | meta y | end line
line ell big c because 1rule deduction modus ponens ell i indeed
  meta x <= 0 imply 0 <= meta y imply | meta x + meta y | <= | meta x | + | meta y | end line
line ell big d because 1rule deduction modus ponens ell l indeed
  meta x <= 0 imply meta y <= 0 imply | meta x + meta y | <= | meta x | + | meta y | end line

line ell big e because lemma from leqGeq modus ponens ell big a modus ponens ell big c indeed
  0 <= meta y imply | meta x + meta y | <= | meta x | + | meta y | end line
line ell big f because lemma from leqGeq modus ponens ell big b modus ponens ell big d indeed
  meta y <= 0 imply | meta x + meta y | <= | meta x | + | meta y | end line

because lemma from leqGeq modus ponens ell big e modus ponens ell big f indeed  
| meta x + meta y | <= | meta x | + | meta y | qed
end math ]"

"[  math in theory system Q lemma lemma insertMiddleTerm(Sum) says
for all terms meta x comma meta y comma meta z indeed
meta x + meta y = parenthesis meta x - meta z end parenthesis + parenthesis meta z + meta y end parenthesis
end lemma end math ]"

"[ math system Q proof of lemma insertMiddleTerm(Sum) reads
any term meta x comma meta y comma meta z end line
line ell a because lemma x=x+y-y indeed meta x = meta x + meta z - meta z end line
line ell b because lemma three2threeTerms indeed 
  meta x + meta z - meta z = meta x - meta z + meta z end line
line ell c because lemma eqTransitivity modus ponens ell a modus ponens ell b indeed
  meta x = meta x - meta z + meta z end line

line ell d because lemma eqAddition modus ponens ell c indeed
  meta x + meta y = parenthesis meta x - meta z end parenthesis + meta z + meta y end line
line ell e because axiom plusAssociativity indeed
  parenthesis meta x - meta z end parenthesis + meta z + meta y  =
  parenthesis meta x - meta z end parenthesis + parenthesis meta z + meta y end parenthesis end line

because lemma eqTransitivity modus ponens ell d modus ponens ell e indeed 
  meta x + meta y = 
  parenthesis meta x - meta z end parenthesis + parenthesis meta z + meta y end parenthesis qed
end math ]"


"[  math in theory system Q lemma lemma insertMiddleTerm(Numerical) says
for all terms meta x comma meta y comma meta z indeed
| meta x + meta y | <= | meta x - meta z | + | meta z + meta y |
end lemma end math ]"

"[ math system Q proof of lemma insertMiddleTerm(Numerical) reads
any term meta x comma meta y comma meta z end line
line ell a because lemma splitNumericalSum indeed
  | parenthesis meta x - meta z end parenthesis + parenthesis meta z + meta y end parenthesis | <=
  | meta x - meta z | + | meta z + meta y | end line

line ell b because lemma insertMiddleTerm(Sum) indeed
  meta x + meta y = parenthesis meta x - meta z end parenthesis + 
                    parenthesis meta z + meta y end parenthesis end line
line ell c because lemma sameNumerical modus ponens ell b indeed
  | meta x + meta y | = | parenthesis meta x - meta z end parenthesis + 
                          parenthesis meta z + meta y end parenthesis | end line
line ell d because lemma eqSymmetry modus ponens ell c indeed
  | parenthesis meta x - meta z end parenthesis + 
    parenthesis meta z + meta y end parenthesis | = | meta x + meta y | end line

because lemma subLeqLeft modus ponens ell d modus ponens ell a indeed  
  | meta x + meta y | <= | meta x - meta z | + | meta z + meta y | qed
end math ]"

(*** REGNESTYKKER ***)



"[ math in theory system Q lemma lemma insertMiddleTerm(Difference) says
for all terms meta x comma meta y comma meta z indeed
meta x - meta y = parenthesis meta x + meta z end parenthesis -
                  parenthesis meta y + meta z end parenthesis
end lemma end math ]"

"[ math system Q proof of lemma insertMiddleTerm(Difference) reads
any term meta x comma meta y comma meta z end line
line ell a because lemma insertMiddleTerm(Sum) indeed
  meta x - meta y = parenthesis meta x - - meta z end parenthesis + 
                    parenthesis - meta z - meta y end parenthesis end line

line ell b because lemma doubleMinus indeed - - meta z = meta z end line
line ell c because lemma eqAdditionLeft modus ponens ell  b indeed
  meta x - - meta z = meta x + meta z end line

line ell d because axiom plusCommutativity indeed - meta z - meta y = - meta y - meta z end line
line ell e because lemma -x-y=-(x+y) indeed
  - meta y - meta z = - parenthesis meta y + meta z end parenthesis end line
line ell f because lemma eqTransitivity modus ponens ell d modus ponens ell e indeed
  - meta z - meta y = - parenthesis meta y + meta z end parenthesis end line

line ell g because lemma addEquations modus ponens ell c modus ponens ell f indeed
  parenthesis meta x - - meta z end parenthesis + parenthesis - meta z - meta y end parenthesis =
  parenthesis meta x +   meta z end parenthesis - parenthesis   meta y + meta z end parenthesis end line

because lemma eqTransitivity modus ponens ell a modus ponens ell g indeed 
  meta x - meta y = parenthesis meta x + meta z end parenthesis -
                    parenthesis meta y + meta z end parenthesis qed
end math ]"



"[  math in theory system Q lemma lemma x+x=2*x says
for all terms meta x indeed
meta x + meta x = 2 * meta x 
end lemma end math ]"


"[ math system Q proof of lemma x+x=2*x reads
any term meta x end line

line ell a because axiom times1    indeed meta x * 1 = meta x end line
line ell b because lemma eqSymmetry indeed meta x = meta x * 1 end line
line ell c because lemma eqAdditionLeft modus ponens ell b indeed 
  meta x + meta x = meta x + meta x  * 1 end line
line ell d because lemma eqAddition modus ponens ell b indeed 
  meta x + meta x * 1 = meta x * 1 + meta x * 1 end line 
line ell e because lemma eqTransitivity modus ponens ell c modus ponens ell d indeed 
  meta x + meta x = meta x * 1 + meta x * 1 end line

line ell f because lemma distributionOut indeed 
  meta x * 1 + meta x * 1 = meta x * parenthesis 1 + 1 end parenthesis end line 
line ell g because 1rule repetition modus ponens ell f indeed ""; XXREP 2 
  meta x * 1 + meta x * 1 = meta x * 2 end line

line ell h because axiom timesCommutativity indeed meta x * 2 = 2 * meta x end line

because lemma eqTransitivity4 modus ponens ell e modus ponens ell g modus ponens ell h indeed  
  meta x + meta x = 2 * meta x qed
end math ]"


"[  math in theory system Q lemma lemma (1/2)x+(1/2)x=x says
for all terms meta x indeed
1/2 * meta x + 1/2 * meta x = meta x
end lemma end math ]"

"[ math system Q proof of lemma (1/2)x+(1/2)x=x reads
any term meta x end line
line ell a because lemma 0<2 indeed 0 < 2 end line
line ell b because lemma lessNeq modus ponens ell a indeed  0 != 2 end line
line ell c because lemma neqSymmetry modus ponens ell b indeed 2 != 0 end line
line ell d because lemma x+x=2*x indeed
  1/2 * meta x + 1/2 * meta x = 2 * parenthesis 1/2 * meta x end parenthesis end line

line ell e because axiom timesAssociativity indeed
  2 * 1/2 * meta x = 2 * parenthesis 1/2 * meta x end parenthesis end line
line ell big a because lemma eqSymmetry modus ponens ell e indeed
  2 * parenthesis 1/2 * meta x end parenthesis = 2 * 1/2 * meta x end line

line ell g because lemma reciprocal modus ponens ell c indeed 2 * 1/2 = 1 end line
line ell h because lemma eqMultiplication modus ponens ell g indeed 
  2 * 1/2 * meta x = 1 * meta x end line

line ell i because lemma times1Left indeed 1 * meta x = meta x end line

because lemma eqTransitivity5 modus ponens ell d modus ponens ell big a 
  modus ponens ell h modus ponens ell i indeed 1/2 * meta x + 1/2 * meta x = meta x qed

end math ]"

"[  math in theory system Q lemma lemma x+x+x=3*x says
for all terms meta x indeed
meta x + meta x + meta x = 3 * meta x
end lemma end math ]"

"[ math system Q proof of lemma x+x+x=3*x reads
any term meta x end line
line ell a because lemma x+x=2*x indeed meta x + meta x = 2 * meta x end line
line ell b because lemma times1Left indeed 1 * meta x = meta x end line
line ell c because lemma eqSymmetry modus ponens ell b indeed meta x = 1 * meta x end line
line ell d because lemma addEquations modus ponens ell a modus ponens ell c indeed
  meta x + meta x + meta x = 2 * meta x + 1 * meta x end line
line ell e because lemma distributionOutLeft indeed 2 * meta x + 1 * meta x = 
  meta x * parenthesis 2 + 1 end parenthesis end line
line ell f because axiom timesCommutativity indeed
  meta x * parenthesis 2 + 1 end parenthesis = parenthesis 2 + 1 end parenthesis * meta x end line
line ell g because lemma eqTransitivity4 modus ponens ell d modus ponens ell e modus ponens ell f indeed
  meta x + meta x + meta x = parenthesis 2 + 1 end parenthesis * meta x end line

because 1rule repetition modus ponens ell g indeed meta x + meta x + meta x = 3 * meta x qed ""; XXREP 3
end math ]"

"[  math in theory system Q lemma lemma 0<3 says 0 < 3
end lemma end math ]"

"[ math system Q proof of lemma 0<3 reads
line ell a because lemma 0<2 indeed 0 < 2 end line
line ell b because lemma lessLeq modus ponens ell a indeed 0 <= 2 end line
line ell c because lemma leqPlus1 modus ponens ell b indeed 0 < 2 + 1 end line
because 1rule repetition modus ponens ell c indeed 0 < 3  qed ""; XXREP 3
end math ]"

"[  math in theory system Q lemma lemma (1/3)x+(1/3)x+(1/3)x=x says
for all terms meta x indeed
1/3 * meta x + 1/3 * meta x + 1/3 * meta x = meta x 
end lemma end math ]"

"[ math system Q proof of lemma (1/3)x+(1/3)x+(1/3)x=x reads
""; XX strukturen er identisk med (1/2)x+(1/2)x=x - maaske mulighed for optimering
any term meta x end line
line ell a because lemma 0<3 indeed 0 < 3 end line
line ell b because lemma positiveNonzero modus ponens ell a indeed 3 != 0 end line
line ell d because lemma x+x+x=3*x indeed
  1/3 * meta x + 1/3 * meta x + 1/3 * meta x = 3 * parenthesis 1/3 * meta x end parenthesis end line

line ell e because axiom timesAssociativity indeed
  3 * 1/3 * meta x = 3 * parenthesis 1/3 * meta x end parenthesis end line
line ell f because lemma eqSymmetry modus ponens ell e indeed
  3 * parenthesis 1/3 * meta x end parenthesis = 3 * 1/3 * meta x end line

line ell g because lemma reciprocal modus ponens ell b indeed 3 * 1/3 = 1 end line
line ell h because lemma eqMultiplication modus ponens ell g indeed 
  3 * 1/3 * meta x = 1 * meta x end line

line ell i because lemma times1Left indeed 1 * meta x = meta x end line

because lemma eqTransitivity5 modus ponens ell d modus ponens ell f modus ponens ell h 
  modus ponens ell i indeed 1/3 * meta x + 1/3 * meta x + 1/3 * meta x = meta x qed
end math ]"

"[  math in theory system Q lemma lemma 0<1/3 says 0 < 1/3
end lemma end math ]"

"[ math system Q proof of lemma 0<1/3 reads
line ell a because lemma 0<3 indeed 0 < 3 end line
because lemma positiveInverted modus ponens ell a indeed 0 < 1/3 qed
end math ]"


"[  math in theory system Q lemma lemma times(-1) says
for all terms meta x indeed
meta x * (-1) = - meta x 
end lemma end math ]"

"[ math system Q proof of lemma times(-1) reads
any term meta x end line
line ell a because axiom negative indeed 1 + (-1) = 0 end line
line ell b because axiom plusCommutativity indeed (-1) + 1 = 1 + (-1) end line
line ell c because lemma eqTransitivity modus ponens ell b modus ponens ell a indeed 
  (-1) + 1 = 0 end line
line ell d because lemma eqMultiplicationLeft modus ponens ell c indeed 
  meta x * parenthesis (-1) + 1 end parenthesis = meta x * 0 end line
line ell e because lemma x*0=0 indeed meta x * 0 = 0 end line
line ell f because lemma eqTransitivity modus ponens ell d modus ponens ell e indeed
  meta x * parenthesis (-1) + 1 end parenthesis = 0 end line

line ell g because axiom distribution indeed 
  meta x * parenthesis (-1) + 1 end parenthesis = meta x * (-1) + meta x * 1 end line
line ell h because lemma eqSymmetry modus ponens ell g indeed
  meta x * (-1) + meta x * 1 =  meta x * parenthesis (-1) + 1 end parenthesis end line

line ell i because lemma eqTransitivity modus ponens ell h modus ponens ell f indeed
  meta x * (-1) + meta x * 1 = 0 end line
line ell j because lemma positiveToRight(Eq) modus ponens ell i indeed
  meta x * (-1) = 0 - parenthesis meta x * 1 end parenthesis end line

line ell k because lemma plus0Left indeed 
  0 - parenthesis meta x * 1 end parenthesis = - parenthesis meta x * 1 end parenthesis end line

line ell m because lemma eqTransitivity modus ponens ell j modus ponens ell k indeed
  meta x * (-1) = - parenthesis meta x * 1 end parenthesis end line

line ell n because axiom times1 indeed meta x * 1 = meta x end line
line ell o because lemma eqNegated modus ponens ell n indeed 
  - parenthesis meta x * 1 end parenthesis = - meta x end line

because lemma eqTransitivity modus ponens ell m modus ponens ell o indeed  
  meta x * (-1) = - meta x qed
end math ]"

"[  math in theory system Q lemma lemma times(-1)Left says
for all terms meta x indeed
(-1) * meta x = - meta x
end lemma end math ]"

"[ math system Q proof of lemma times(-1)Left reads
any term meta x end line
line ell a because lemma times(-1) indeed meta x * (-1) = - meta x end line
line ell b because axiom timesCommutativity indeed (-1) * meta x = meta x * (-1) end line
because lemma eqTransitivity modus ponens ell b modus ponens ell a indeed (-1) * meta x = - meta x qed
end math ]"







(*** KVANTI ***)

\begin{list}{}{
\setlength{\leftmargin}{5em}
\setlength{\itemindent}{-5em}}

\item "[ math macro define Nat( var x ) as 
  lambda var c dot quote var x end quote term in 
    parenthesis quote meta v2n end quote pair quote meta m end quote pair quote meta n end quote pair true 
    end parenthesis
end define end math ]"

\item "[ math macro define meta-sub var a is var b where var x is var t end sub as 
  meta-sub1 quote var a end quote is quote var b end quote 
  where quote var x end quote is quote var t end   quote end sub 
end define end math ]"

\item "[ math value define meta-sub1  var a is var b where var x is var t end sub as var a tagged guard var x tagged guard var t tagged guard newline

open if var b term root equal quote for all var u indeed var v end quote macro and var b first term equal var x then var a term equal var b else newline

open if var b term equal var x then var a term equal var t else newline

var a term root equal var b macro and meta-sub* var a tail is var b tail where var x is var t end sub
end define end math ]"

\item "[ math value define meta-sub* var a is var b where var x is var t end sub as 
var b tagged guard var x tagged guard var t tagged guard 
tagged if var a then true else 
meta-sub1 var a head is var b head where var x is var t end sub macro and 
meta-sub* var a tail is var b tail where var x is var t end sub end if 
end define end math ]"

\end{list}




"[  math in theory system Q lemma pred lemma allNegated(Imply) says
for all terms meta v1 comma meta a indeed
not0 for all meta v1 indeed meta a imply
exist0 meta v1 indeed not0 meta a 
end lemma end math ]"

"[ math system Q proof of pred lemma allNegated(Imply) reads
block any term meta v1 comma meta a end line
line ell a premise for all meta v1 indeed not0 not0 meta a end line
""; XX jeg kan bedre lide - at meta v1 - i den foelgende linie
line ell b because lemma a4 at meta x modus ponens ell a indeed not0 not0 meta a end line
line ell c because prop lemma remove double neg modus ponens ell b indeed meta a end line
because 1rule gen modus ponens ell c indeed for all meta v1 indeed meta a end line
line ell d end block

any term meta v1 comma meta a end line
line ell e because 1rule deduction modus ponens ell d indeed
  for all meta v1 indeed not0 not0 meta a imply for all meta v1 indeed meta a end line

line ell f because prop lemma contrapositive modus ponens ell e indeed
  not0 for all meta v1 indeed meta a imply not0 for all meta v1 indeed not0 not0 meta a end line

because 1rule repetition modus ponens ell f indeed ""; XXREP exist
  not0 for all meta v1 indeed meta a imply exist0 meta v1 indeed not0 meta a qed
end math ]"


"[  math in theory system Q lemma pred lemma existNegated(Imply) says
for all terms meta v1 comma meta a indeed
not0 exist0 meta v1 indeed meta a imply
for all meta v1 indeed not0 meta a
end lemma end math ]"

"[ math system Q proof of pred lemma existNegated(Imply) reads
block any term meta v1 comma meta a end line
line ell a premise not0 exist0 meta v1 indeed meta a end line
line ell b because 1rule repetition modus ponens ell a indeed ""; XXREP exist
  not0 not0 for all meta v1 indeed not0 meta a end line
because prop lemma remove double neg modus ponens ell b indeed 
  for all meta v1 indeed not0 meta a end line
line ell c end block 

any term meta v1 comma meta a end line
because 1rule deduction modus ponens ell c indeed 
  not0 exist0 meta v1 indeed meta a imply for all meta v1 indeed not0 meta a qed

end math ]"

"[  math in theory system Q lemma pred lemma addAll says
for all terms meta v1 comma meta a comma meta b indeed
meta a imply meta b infer 
for all meta v1 indeed meta a imply for all meta v1 indeed meta b
end lemma end math ]"

"[ math system Q proof of pred lemma addAll reads
block any term meta v1 comma meta a comma meta b end line
line ell a premise meta a imply meta b end line
line ell b premise for all meta v1 indeed meta a end line
line ell c because lemma a4 modus ponens ell b indeed meta a end line
line ell d because 1rule mp modus ponens ell a modus ponens ell c indeed meta b end line
because 1rule gen modus ponens ell d indeed for all meta v1 indeed meta b end line
line ell big x end block

any term meta v1 comma meta a comma meta b end line
line ell a because 1rule deduction modus ponens ell big x indeed 
  parenthesis meta a imply meta b end parenthesis imply 
  for all meta v1 indeed meta a imply for all meta v1 indeed meta b end line

line ell b premise meta a imply meta b end line
because 1rule mp modus ponens ell a modus ponens ell b indeed 
  for all meta v1 indeed meta a imply for all meta v1 indeed meta b qed
end math ]"

"[  math in theory system Q lemma pred lemma addExist helper1 says
for all terms meta y comma meta v1 comma meta v2 comma meta a comma meta b comma meta c comma meta d indeed
meta-sub not0 meta b is not0 meta d where meta v2 is meta y end sub endorse
parenthesis meta a imply meta b end parenthesis imply
parenthesis meta c imply meta a end parenthesis imply
exist0 meta v1 indeed meta c imply  for all meta v2 indeed not0 meta d imply
not0 for all meta v2 indeed not0 meta d 
end lemma end math ]"

"[ math system Q proof of pred lemma addExist helper1 reads

block any term meta y comma meta v1 comma meta v2 


      comma meta a comma meta b comma meta c comma meta d end line
line ell big x side condition meta-sub not0 meta b is not0 meta d where meta v2 is meta y end sub end line
line ell a premise meta a imply meta b end line
line ell big x premise meta c imply meta a end line
line ell b premise exist0 meta v1 indeed meta c end line
line ell c premise for all meta v2 indeed not0 meta d end line
line ell d because lemma a4 at meta y modus ponens ell c indeed not0 meta b end line
line ell e because prop lemma mt modus ponens ell a modus ponens ell d indeed not0 meta a end line
line ell big y because prop lemma mt modus ponens ell big x modus ponens ell e indeed not0 meta c end line
line ell f because 1rule gen modus ponens ell big y indeed 
  for all meta v1 indeed not0 meta c end line
line ell g because 1rule repetition modus ponens ell b indeed ""; XXREP exist
  not0 for all meta v1 indeed not0 meta c end line
because prop lemma from contradiction modus ponens ell f modus ponens ell g indeed
  not0 for all meta v2 indeed not0 meta d end line
line ell big a end block

any term meta y comma meta v1 comma meta v2 comma meta a comma meta b comma meta c comma meta d end line
because 1rule deduction modus ponens ell big a indeed
  meta-sub not0 meta b is not0 meta d where meta v2 is meta y end sub endorse
  parenthesis meta a imply meta b end parenthesis imply
  parenthesis meta c imply meta a end parenthesis imply
  exist0 meta v1 indeed meta c imply  for all meta v2 indeed not0 meta d imply
  not0 for all meta v2 indeed not0 meta d qed
end math ]"


"[  math in theory system Q lemma pred lemma addExist helper2 says
for all terms meta y comma meta v1 comma meta v2 comma meta a comma meta b comma meta c comma meta d indeed
meta-sub not0 meta b is not0 meta d where meta v2 is meta y end sub endorse
parenthesis meta a imply meta b end parenthesis imply
parenthesis meta c imply meta a end parenthesis imply
exist0 meta v1 indeed meta c imply
exist0 meta v2 indeed meta d 
end lemma end math ]"

"[ math system Q proof of pred lemma addExist helper2 reads
block any term meta y comma meta v1 comma meta v2 
      comma meta a comma meta b comma meta c comma meta d end line
line ell b side condition meta-sub not0 meta b is not0 meta d where meta v2 is meta y end sub end line
line ell c premise meta a imply meta b end line
line ell big x premise meta c imply meta a end line
line ell d premise exist0 meta v1 indeed meta c end line

line ell e because pred lemma addExist helper1 modus probans ell b indeed
  parenthesis meta a imply meta b end parenthesis imply
  parenthesis meta c imply meta a end parenthesis imply
  exist0 meta v1 indeed meta c imply  for all meta v2 indeed not0 meta d imply
  not0 for all meta v2 indeed not0 meta d end line

line ell f because prop lemma mp3 modus ponens ell e modus ponens ell c modus ponens ell big x
  modus ponens ell d indeed 
  for all meta v2 indeed not0 meta d imply
  not0 for all meta v2 indeed not0 meta d end line

line ell g because prop lemma imply negation modus ponens ell f indeed
  not0 for all meta v2 indeed not0 meta d end line

because 1rule repetition modus ponens ell g indeed exist0 meta v2 indeed meta d end line ""; XXREP exist
line ell big a end block

any term meta y comma meta v1 comma meta v2 comma meta a comma meta b comma meta c comma meta d end line

because 1rule deduction modus ponens ell big a indeed
 meta-sub not0 meta b is not0 meta d where meta v2 is meta y end sub endorse
 parenthesis meta a imply meta b end parenthesis imply
 parenthesis meta c imply meta a end parenthesis imply
 exist0 meta v1 indeed meta c imply
 exist0 meta v2 indeed meta d qed
end math ]"

"[  math in theory system Q lemma pred lemma addExist says

for all terms meta y comma meta v1 comma meta v2 comma meta a comma meta b comma meta c comma meta d indeed
meta-sub not0 meta b is not0 meta d where meta v2 is meta y end sub endorse



meta a imply meta b infer
meta c imply meta a infer
exist0 meta v1 indeed meta c imply exist0 meta v2 indeed meta d
end lemma end math ]"

"[ math system Q proof of pred lemma addExist reads

any term meta y comma meta v1 comma meta v2 comma meta a comma meta b comma meta c comma meta d end line

line ell b side condition meta-sub not0 meta b is not0 meta d where meta v2 is meta y end sub end line
line ell c premise meta a imply meta b end line
line ell big x premise meta c imply meta a end line

line ell d because pred lemma addExist helper2 modus probans ell b indeed
  parenthesis meta a imply meta b end parenthesis imply
  parenthesis meta c imply meta a end parenthesis imply
  exist0 meta v1 indeed meta c imply exist0 meta v2 indeed meta d end line

because prop lemma mp2 modus ponens ell d modus ponens ell c modus ponens ell big x indeed
  exist0 meta v1 indeed meta c imply exist0 meta v2 indeed meta d qed
end math ]"


"[  math in theory system Q lemma pred lemma addExist(SimpleAnt) says
for all terms meta y comma meta v1 comma meta v2 comma meta a comma meta b comma meta d indeed
meta-sub not0 meta b is not0 meta d where meta v2 is meta y end sub endorse

meta a imply meta b infer
exist0 meta v1 indeed meta a imply exist0 meta v2 indeed meta d
end lemma end math ]"


"[ math system Q proof of pred lemma addExist(SimpleAnt) reads
for all terms meta y comma meta v1 comma meta v2 comma meta a comma meta b comma meta d indeed
line ell a side condition meta-sub not0 meta b is not0 meta d where meta v2 is meta y end sub end line
line ell b premise meta a imply meta b end line
line ell c because prop lemma auto imply indeed meta a imply meta a end line
because pred lemma addExist at meta y modus probans ell a modus ponens ell b modus ponens ell c indeed 
  exist0 meta v1 indeed meta a imply exist0 meta v2 indeed meta d qed
end math ]"


"[  math in theory system Q lemma pred lemma addExist(Simple) says
for all terms meta v1 comma meta v2 comma meta a comma meta b indeed
meta a imply meta b infer
exist0 meta v1 indeed meta a imply exist0 meta v2 indeed meta b
end lemma end math ]"

"[ math system Q proof of pred lemma addExist(Simple) reads
any term meta v1 comma meta v2 comma meta a comma meta b end line
line ell a premise meta a imply meta b end line
line ell b because prop lemma auto imply indeed meta a imply meta a end line

because pred lemma addExist at meta v2 modus ponens ell a modus ponens ell b indeed 
  exist0 meta v1 indeed meta a imply exist0 meta v2 indeed meta b qed
end math ]"

"[  math in theory system Q lemma pred lemma AEAnegated says
for all terms meta v1 comma meta v2 comma meta v3 comma meta a indeed
not0 for all meta v1 indeed exist0 meta v2 indeed for all meta v3 indeed meta a infer
exist0 meta v1 indeed for all meta v2 indeed exist0 meta v3 indeed not0 meta a
end lemma end math ]"

"[ math system Q proof of pred lemma AEAnegated reads
any term meta v1 comma meta v2 comma meta v3 comma meta a end line

line ell big a premise 
  not0 for all meta v1 indeed exist0 meta v2 indeed for all meta v3 indeed meta a end line

line ell a because pred lemma allNegated(Imply) indeed
  not0 for all meta v3 indeed meta a imply exist0 meta v3 indeed not0 meta a end line

line ell b because pred lemma addAll modus ponens ell a indeed
  for all meta v2 indeed not0 for all meta v3 indeed meta a imply 
  for all meta v2 indeed exist0 meta v3 indeed not0 meta a end line
line ell c because pred lemma existNegated(Imply) indeed
  not0 exist0 meta v2 indeed for all meta v3 indeed meta a imply
  for all meta v2 indeed not0 for all meta v3 indeed meta a end line
line ell d because prop lemma imply transitivity modus ponens ell c modus ponens ell b indeed
  not0 exist0 meta v2 indeed for all meta v3 indeed meta a imply
  for all meta v2 indeed exist0 meta v3 indeed not0 meta a end line

line ell e because pred lemma addExist(Simple) modus ponens ell d indeed
  exist0 meta v1 indeed not0 exist0 meta v2 indeed for all meta v3 indeed meta a imply
  exist0 meta v1 indeed for all meta v2 indeed exist0 meta v3 indeed not0 meta a end line
line ell f because pred lemma allNegated(Imply) indeed
  not0 for all meta v1 indeed exist0 meta v2 indeed for all meta v3 indeed meta a imply
  exist0 meta v1 indeed not0 exist0 meta v2 indeed for all meta v3 indeed meta a end line
line ell g because prop lemma imply transitivity modus ponens ell f modus ponens ell e indeed 
  not0 for all meta v1 indeed exist0 meta v2 indeed for all meta v3 indeed meta a imply
  exist0 meta v1 indeed for all meta v2 indeed exist0 meta v3 indeed not0 meta a end line

because 1rule mp modus ponens ell g modus ponens ell big a indeed
  exist0 meta v1 indeed for all meta v2 indeed exist0 meta v3 indeed not0 meta a qed
end math ]"

"[  math in theory system Q lemma pred lemma addEAE says
for all terms meta v1 comma meta v2 comma meta v3 comma meta a comma meta b indeed
meta a imply meta b infer
exist0 meta v1 indeed for all meta v2 indeed exist0 meta v3 indeed meta a imply
exist0 meta v1 indeed for all meta v2 indeed exist0 meta v3 indeed meta b
end lemma end math ]"

"[ math system Q proof of pred lemma addEAE reads
any term meta v1 comma meta v2 comma meta v3 comma meta a comma meta b end line
line ell a premise meta a imply meta b end line

line ell b because pred lemma addExist(Simple) modus ponens ell a indeed
  exist0 meta v3 indeed meta a imply exist0 meta v3 indeed meta b end line
line ell c because pred lemma addAll   modus ponens ell b indeed
  for all meta v2 indeed exist0 meta v3 indeed meta a imply 
  for all meta v2 indeed exist0 meta v3 indeed meta b end line
because pred lemma addExist(Simple) modus ponens ell c indeed
  exist0 meta v1 indeed for all meta v2 indeed exist0 meta v3 indeed meta a imply 
  exist0 meta v1 indeed for all meta v2 indeed exist0 meta v3 indeed meta b qed
end math ]"



"[  math in theory system Q lemma lemma fromNotSameF(Weak)(Helper) says
for all terms meta x comma meta y comma meta z indeed
not0 | meta x - meta y | < meta z infer
meta x <= meta y - meta z or0 meta y <= meta x - meta z
end lemma end math ]"

"[ math system Q proof of lemma fromNotSameF(Weak)(Helper) reads
block any term meta x comma meta y comma meta z end line
line ell a premise meta z <= | meta x - meta y | end line
line ell b premise 0 <= meta x - meta y end line
line ell c because lemma nonnegativeNumerical modus ponens ell b indeed
  | meta x - meta y | = meta x - meta y end line
line ell d because lemma subLeqRight modus ponens ell c modus ponens ell a indeed
  meta z <= meta x - meta y end line

line ell e because lemma negativeToLeft(Leq) modus ponens ell d indeed
  meta z + meta y <= meta x end line
line ell f because axiom plusCommutativity indeed meta z + meta y = meta y + meta z end line
line ell g because lemma subLeqLeft modus ponens ell f modus ponens ell e indeed
  meta y + meta z <= meta x end line

line ell h because lemma positiveToRight(Leq) modus ponens ell g indeed meta y <= meta x - meta z end line
because prop lemma weaken or first modus ponens ell h indeed 
  meta x <= meta y - meta z or0 meta y <= meta x - meta z end line
line ell big a end block

block any term meta x comma meta y comma meta z end line
line ell a premise meta z <= | meta x - meta y | end line
line ell b premise not0 0 <= meta x - meta y end line
line ell c because lemma toLess modus ponens ell b indeed meta x - meta y < 0 end line
line ell d because lemma negativeNumerical indeed 
  | meta x - meta y | = - parenthesis meta x - meta y end parenthesis end line
line ell e because lemma minusNegated indeed 
  - parenthesis meta x - meta y end parenthesis = meta y - meta x end line
line ell f because lemma eqTransitivity modus ponens ell d modus ponens ell e indeed
  | meta x - meta y | = meta y - meta x end line
line ell g because lemma subLeqRight modus ponens ell f modus ponens ell a indeed
  meta z <= meta y - meta x end line

line ell h because lemma negativeToLeft(Leq) modus ponens ell g indeed
  meta z + meta x <= meta y end line
line ell i because axiom plusCommutativity indeed meta z + meta x = meta x + meta z end line
line ell j because lemma subLeqLeft modus ponens ell i modus ponens ell h indeed
  meta x + meta z <= meta y end line
line ell k because lemma positiveToRight(Leq) modus ponens ell j indeed
  meta x <= meta y - meta z end line

because prop lemma weaken or second modus ponens ell k indeed
  meta x <= meta y - meta z or0 meta y <= meta x - meta z end line
line ell big b end block

any term meta x comma meta y comma meta z end line
line ell a because 1rule deduction modus ponens ell big a indeed
  meta z <= | meta x - meta y | imply 0 <= meta x - meta y imply
  meta x <= meta y - meta z or0 meta y <= meta x - meta z end line
  
line ell b because 1rule deduction modus ponens ell big b indeed
  meta z <= | meta x - meta y | imply not0 0 <= meta x - meta y imply
  meta x <= meta y - meta z or0 meta y <= meta x - meta z end line

line ell c premise not0 | meta x - meta y | < meta z end line
line ell d because lemma fromNotLess modus ponens ell c indeed
  meta z <= | meta x - meta y | end line

line ell e because 1rule mp modus ponens ell a modus ponens ell d indeed
  0 <= meta x - meta y imply meta x <= meta y - meta z or0 meta y <= meta x - meta z end line
line ell f because 1rule mp modus ponens ell b modus ponens ell d indeed


  not0 0 <= meta x - meta y imply meta x <= meta y - meta z or0 meta y <= meta x - meta z end line

because prop lemma from negations modus ponens ell e modus ponens ell f indeed  
  meta x <= meta y - meta z or0 meta y <= meta x - meta z qed
end math ]"

"[  math in theory system Q lemma lemma fromNotSameF(Weak) says
for all terms meta m comma meta n comma meta ep comma meta fx comma meta fy indeed
not0 meta fx sameF meta fy infer
exist0 meta ep indeed for all meta n indeed exist0 meta m indeed 

0 < meta ep and0 meta n <= meta m and0 
parenthesis [ meta fx ; meta m ] <= [ meta fy ; meta m ] - meta ep or0 
[ meta fy ; meta m ] <= [ meta fx ; meta m ] - meta ep end parenthesis 
end lemma end math ]"

"[ math system Q proof of lemma fromNotSameF(Weak) reads
any term meta m comma meta n comma meta ep comma meta fx comma meta fy end line
line ell a premise not0 meta fx sameF meta fy end line
line ell x because 1rule repetition modus ponens ell a indeed ""; XXREP sameF
  not0 for all object ep indeed exist0 object n indeed for all object m indeed ( 0 < object ep imply 
  object n <= object m imply | [ meta fx ; object m ] - [ meta fy ; object m ] | < object ep ) end line
line ell b because 1rule deduction modus ponens ell x indeed
  not0 for all meta ep indeed exist0 meta n indeed for all meta m indeed parenthesis
  0 < meta ep imply meta n <= meta m imply | [ meta fx ; meta m ] - [ meta fy ; meta m ] | < meta ep 
  end parenthesis end line

line ell c because pred lemma AEAnegated modus ponens ell b indeed
  exist0 meta ep indeed for all meta n indeed exist0 meta m indeed  
  not0 parenthesis 0 < meta ep imply 
  meta n <= meta m imply | [ meta fx ; meta m ] - [ meta fy ; meta m ] | < meta ep end parenthesis end line

block any term meta m comma meta n comma meta ep comma meta fx comma meta fy end line
line ell a premise not0 parenthesis 0 < meta ep imply 
  meta n <= meta m imply | [ meta fx ; meta m ] - [ meta fy ; meta m ] | < meta ep end parenthesis end line
line ell b because prop lemma from negated double imply modus ponens ell a indeed
  0 < meta ep and0 meta n <= meta m and0 
  not0 | [ meta fx ; meta m ] - [ meta fy ; meta m ] | < meta ep end line
line ell c because prop lemma first conjunct modus ponens ell b indeed
  0 < meta ep and0 meta n <= meta m end line
line ell d because prop lemma second conjunct modus ponens ell b indeed
  not0 | [ meta fx ; meta m ] - [ meta fy ; meta m ] | < meta ep end line
line ell g because lemma fromNotSameF(Weak)(Helper) modus ponens ell d indeed
  [ meta fx ; meta m ] <= [ meta fy ; meta m ] - meta ep or0 
  [ meta fy ; meta m ] <= [ meta fx ; meta m ] - meta ep end line
because prop lemma join conjuncts modus ponens ell c modus ponens ell g indeed
  0 < meta ep and0 meta n <= meta m and0 
  parenthesis  [ meta fx ; meta m ] <= [ meta fy ; meta m ] - meta ep or0 
  [ meta fy ; meta m ] <= [ meta fx ; meta m ] - meta ep end parenthesis end line
line ell big a end block

line ell d because 1rule deduction modus ponens ell big a indeed not0 parenthesis 0 < meta ep imply
  meta n <= meta m imply | [ meta fx ; meta m ] - [ meta fy ; meta m ] | < meta ep end parenthesis imply
  0 < meta ep and0 meta n <= meta m and0 
  parenthesis [ meta fx ; meta m ] <= [ meta fy ; meta m ] - meta ep or0 
  [ meta fy ; meta m ] <= [ meta fx ; meta m ] - meta ep end parenthesis end line

line ell e because pred lemma addEAE modus ponens ell d indeed
  exist0 meta ep indeed for all meta n indeed exist0 meta m indeed not0 parenthesis 0 < meta ep imply
  meta n <= meta m imply | [ meta fx ; meta m ] - [ meta fy ; meta m ] | < meta ep end parenthesis imply
  exist0 meta ep indeed for all meta n indeed exist0 meta m indeed
  0 < meta ep and0 meta n <= meta m and0 
  parenthesis [ meta fx ; meta m ] <= [ meta fy ; meta m ] - meta ep or0 
  [ meta fy ; meta m ] <= [ meta fx ; meta m ] - meta ep end parenthesis end line

because 1rule mp modus ponens ell e modus ponens ell c indeed  
  exist0 meta ep indeed for all meta n indeed exist0 meta m indeed
  0 < meta ep and0 meta n <= meta m and0 parenthesis 
  [ meta fx ; meta m ] <= [ meta fy ; meta m ] - meta ep or0 
  [ meta fy ; meta m ] <= [ meta fx ; meta m ] - meta ep end parenthesis qed
end math ]"


"[ math in theory system Q lemma lemma fromMax(1) says
for all terms meta x comma meta y indeed
meta y <= meta x infer max( meta x , meta y ) = meta x
end lemma end math ]"

"[ math system Q proof of lemma fromMax(1) reads
any term meta x comma meta y end line
line ell a premise meta y <= meta x end line
line ell b because axiom max indeed
  ( meta y <= meta x      and0 max( meta x , meta y ) = meta x ) or0 
  ( not0 meta y <= meta x and0 max( meta x , meta y ) = meta y ) end line
line ell c because prop lemma add double neg modus ponens ell a indeed 
  not0 not0 meta y <= meta x end line
line ell d because prop lemma to negated and(1) modus ponens ell c indeed
  not0 ( not0 meta y <= meta x and0 max( meta x , meta y ) = meta y ) end line
line ell e because prop lemma negate second disjunct modus ponens ell b modus ponens ell d indeed
  meta y <= meta x and0 max( meta x , meta y ) = meta x end line
because prop lemma second conjunct modus ponens ell e indeed
  max( meta x , meta y ) = meta x qed
end math ]"

"[ math in theory system Q lemma lemma fromMax(2) says
for all terms meta x comma meta y indeed
not0 meta y <= meta x infer max( meta x , meta y ) = meta y
end lemma end math ]"

"[ math system Q proof of lemma fromMax(2) reads
any term meta x comma meta y end line
line ell a premise not0 meta y <= meta x end line
line ell b because axiom max indeed
  ( meta y <= meta x      and0 max( meta x , meta y ) = meta x ) or0 
  ( not0 meta y <= meta x and0 max( meta x , meta y ) = meta y ) end line
line ell c because prop lemma to negated and(1) modus ponens ell a indeed
  not0 ( meta y <= meta x      and0 max( meta x , meta y ) = meta x ) end line
line ell d because prop lemma negate first disjunct modus ponens ell b modus ponens ell c indeed
  not0 meta y <= meta x and0 max( meta x , meta y ) = meta y end line
because prop lemma second conjunct modus ponens ell d indeed
  max( meta x , meta y ) = meta y qed
end math ]"



"[  math in theory system Q lemma lemma leqMax1 says
for all terms meta x comma meta y indeed
meta x <= max( meta x , meta y )
end lemma end math ]"

"[ math system Q proof of lemma leqMax1 reads
block any term meta x comma meta y end line
line ell a premise meta y <= meta x end line
line ell f because lemma fromMax(1) modus ponens ell a indeed
  max( meta x , meta y ) = meta x end line
line ell g because lemma eqSymmetry modus ponens ell f indeed
  meta x = max( meta x , meta y ) end line
because lemma eqLeq modus ponens ell g indeed meta x <= max( meta x , meta y ) end line
line ell big a end block

block any term meta x comma meta y end line
line ell a premise not0 meta y <= meta x end line
line ell e because lemma fromMax(2) modus ponens ell a indeed
  max( meta x , meta y ) = meta y end line
line ell f because lemma eqSymmetry modus ponens ell e indeed
  meta y = max( meta x , meta y ) end line
line ell g because lemma toLess modus ponens ell a indeed
  meta x < meta y end line
line ell h because lemma lessLeq modus ponens ell g indeed
  meta x <= meta y end line
because lemma subLeqRight modus ponens ell f modus ponens ell h indeed 
  meta x <= max( meta x , meta y ) end line
line ell big b end block

any term meta x comma meta y end line

line ell a because 1rule deduction modus ponens ell big a indeed 
  meta y <= meta x imply meta x <= max( meta x , meta y ) end line

line ell b because 1rule deduction modus ponens ell big b indeed 
  not0 meta y <= meta x imply meta x <= max( meta x , meta y ) end line

because prop lemma from negations modus ponens ell a modus ponens ell b indeed 
  meta x <= max( meta x , meta y ) qed
end math ]"



"[  math in theory system Q lemma lemma leqMax2 says
for all terms meta x comma meta y indeed
meta y <= max( meta x , meta y )
end lemma end math ]"

"[ math system Q proof of lemma leqMax2 reads
block any term meta x comma meta y end line
line ell a premise meta y <= meta x end line
line ell c because lemma fromMax(1) modus ponens ell a indeed
  max( meta x , meta y ) = meta x end line
line ell d because lemma eqSymmetry modus ponens ell c indeed
  meta x = max( meta x , meta y ) end line
because lemma subLeqRight modus ponens ell d modus ponens ell a indeed
  meta y <= max( meta x , meta y ) end line
line ell big a end block

block any term meta x comma meta y end line
line ell a premise not0 meta y <= meta x end line
line ell c because lemma fromMax(2) modus ponens ell a indeed
  max( meta x , meta y ) = meta y end line
line ell d because lemma eqSymmetry modus ponens ell c indeed
  meta y = max( meta x , meta y ) end line
because lemma eqLeq modus ponens ell d indeed 
  meta y <= max( meta x , meta y ) end line
line ell big b end block

any term meta x comma meta y end line

line ell a because 1rule deduction modus ponens ell big a indeed 
  meta y <= meta x imply meta y <= max( meta x , meta y ) end line

line ell b because 1rule deduction modus ponens ell big b indeed 
  not0 meta y <= meta x imply meta y <= max( meta x , meta y ) end line

because prop lemma from negations modus ponens ell a modus ponens ell b indeed 
  meta y <= max( meta x , meta y ) qed
end math ]"






"[  math in theory system Q lemma lemma negativeToRight(Leq) says
for all terms meta x comma meta y comma meta z indeed
meta x - meta y <= meta z infer meta x <= meta z + meta y 
end lemma end math ]"

"[ math system Q proof of lemma negativeToRight(Leq) reads
""; XX hvad med x=x-y+y?
any term meta x comma meta y comma meta z end line
line ell a premise meta x - meta y <= meta z end line
line ell b because lemma leqAddition modus ponens ell a indeed
  meta x - meta y + meta y <= meta z + meta y end line
line ell c because lemma x=x+y-y indeed meta x = meta x + meta y - meta y end line
line ell d because lemma three2threeTerms indeed 
  meta x + meta y - meta y = meta x - meta y + meta y end line
line ell e because lemma eqTransitivity modus ponens ell c modus ponens ell d indeed
  meta x = meta x - meta y + meta y end line
line ell f because lemma eqSymmetry modus ponens ell e indeed meta x - meta y + meta y = meta x end line
because lemma subLeqLeft modus ponens ell f modus ponens ell b indeed 
  meta x <= meta z + meta y qed
end math ]"



"[  math in theory system Q lemma lemma lessThanMax says
for all terms meta a comma meta x comma meta y comma meta z indeed
meta a imply meta x < meta y infer
not0 meta a imply meta x < meta z infer

meta x < max( meta y , meta z )
end lemma end math ]"

"[ math system Q proof of lemma lessThanMax reads

block any term meta a comma meta x comma meta y comma meta z end line
line ell a premise meta a imply meta x < meta y end line
line ell b premise meta a end line
line ell c because 1rule mp modus ponens ell a modus ponens ell b indeed meta x < meta y end line
line ell d because lemma leqMax1 indeed meta y <= max( meta y , meta z ) end line
because lemma lessLeqTransitivity modus ponens ell c modus ponens ell d indeed
  meta x < max( meta y , meta z ) end line
line ell big a end block

block any term meta a comma meta x comma meta y comma meta z end line
line ell a premise not0 meta a imply meta x < meta z end line
line ell b premise not0 meta a end line
line ell c because 1rule mp modus ponens ell a modus ponens ell b indeed meta x < meta z end line
line ell d because lemma leqMax2 indeed meta z <= max( meta y , meta z ) end line
because lemma lessLeqTransitivity modus ponens ell c modus ponens ell d indeed
  meta x < max( meta y , meta z ) end line
line ell big b end block

any term meta a comma meta x comma meta y comma meta z end line

line ell a because 1rule deduction modus ponens ell big a indeed
  parenthesis meta a imply meta x < meta y end parenthesis imply
  meta a imply meta x < max( meta y , meta z ) end line
line ell b because 1rule deduction modus ponens ell big b indeed
  parenthesis not0 meta a imply meta x < meta z end parenthesis imply
  not0 meta a imply meta x < max( meta y , meta z ) end line

line ell c premise meta a imply meta x < meta y end line
line ell d premise not0 meta a imply meta x < meta z end line

line ell e because 1rule mp modus ponens ell a modus ponens ell c indeed
  meta a imply meta x < max( meta y , meta z ) end line
line ell f because 1rule mp modus ponens ell b modus ponens ell d indeed
  not0 meta a imply meta x < max( meta y , meta z ) end line

because prop lemma from negations modus ponens ell e modus ponens ell f indeed 
  meta x < max( meta y , meta z ) qed
end math ]"


"[  math in theory system Q lemma lemma nonnegativeFactors says
for all terms meta x comma meta y indeed
0 <= meta x infer 0 <= meta y infer 0 <= meta x * meta y 
end lemma end math ]"

"[ math system Q proof of lemma nonnegativeFactors reads
any term meta x comma meta y end line
line ell a premise 0 <= meta x end line
line ell b premise 0 <= meta y end line
line ell c because lemma leqMultiplication modus ponens ell b modus ponens ell a indeed
  0 * meta y <= meta x * meta y end line

line ell d because axiom timesCommutativity indeed 0 * meta y = meta y * 0 end line
line ell e because lemma x*0=0 indeed meta y * 0 = 0 end line
line ell f because lemma eqTransitivity modus ponens ell d modus ponens ell e indeed 
  0 * meta y = 0 end line
because lemma subLeqLeft modus ponens ell f modus ponens ell c indeed  0 <= meta x * meta y qed
end math ]"


"[  math in theory system Q lemma lemma multiplyEquations says
for all terms meta x comma meta y comma meta z comma meta u indeed
meta x = meta y infer meta z = meta u infer meta x * meta z = meta y * meta u
end lemma end math ]"

"[ math system Q proof of lemma multiplyEquations reads
any term meta x comma meta y comma meta z comma meta u end line
line ell a premise meta x = meta y end line
line ell b premise meta z = meta u end line
line ell c because lemma eqMultiplication modus ponens ell a indeed 
  meta x * meta z = meta y * meta z end line
line ell d because lemma eqMultiplicationLeft modus ponens ell b indeed
  meta y * meta z = meta y * meta u end line
because lemma eqTransitivity modus ponens ell c modus ponens ell d indeed 
  meta x * meta z = meta y * meta u qed end math ]"




"[  math in theory system Q lemma lemma minusTimesMinus says
for all terms meta x comma meta y indeed
parenthesis - meta x end parenthesis * parenthesis - meta y end parenthesis = meta x * meta y
end lemma end math ]"

"[ math system Q proof of lemma minusTimesMinus reads
any term meta x comma meta y end line
line ell a because lemma doubleMinus indeed - - meta y = meta y end line
line ell b because lemma times(-1)Left indeed 
  (-1) * parenthesis - meta y end parenthesis = - - meta y end line
line ell c because lemma eqTransitivity modus ponens ell b modus ponens ell a indeed
  (-1) * parenthesis - meta y end parenthesis = meta y end line

line ell d because lemma eqMultiplicationLeft modus ponens ell c indeed
  meta x * parenthesis (-1) * parenthesis - meta y end parenthesis end parenthesis = meta x *  meta y end line

line ell e because lemma times(-1) indeed meta x * (-1) = - meta x end line
line ell f because lemma eqMultiplication modus ponens ell e indeed 
  meta x * (-1) * - meta y = - meta x * - meta y end line

line ell g because axiom timesAssociativity indeed 
  meta x * (-1) * - meta y = meta x * parenthesis (-1) * - meta y end parenthesis end line

line ell h because lemma equality modus ponens ell f modus ponens ell g indeed
  - meta x * - meta y = meta x * parenthesis (-1) * - meta y end parenthesis end line

because lemma eqTransitivity modus ponens ell h modus ponens ell d indeed 
  - meta x * - meta y = meta x * meta y qed
end math ]"

"[  math in theory system Q lemma lemma plusTimesMinus says
for all terms meta x comma meta y indeed
meta x * - meta y = - parenthesis meta x * meta y end parenthesis
end lemma end math ]"

"[ math system Q proof of lemma plusTimesMinus reads
any term meta x comma meta y end line
line ell a because lemma times(-1)Left indeed (-1) * meta y = - meta y end line
line ell b because lemma eqMultiplicationLeft modus ponens ell a indeed
  meta x * parenthesis (-1) * meta y end parenthesis = meta x * - meta y end line

line ell c because axiom timesAssociativity indeed 
  meta x * (-1) * meta y = meta x * parenthesis (-1) * meta y end parenthesis end line

line ell d because axiom timesCommutativity indeed meta x * (-1) = (-1) * meta x end line
line ell e because lemma eqMultiplication modus ponens ell d indeed
  meta x * (-1) * meta y = (-1) * meta x * meta y end line
line ell f because axiom timesAssociativity indeed
  (-1) * meta x * meta y = (-1) * parenthesis meta x * meta y end parenthesis end line
line ell g because lemma times(-1)Left indeed
  (-1) * parenthesis meta x * meta y end parenthesis = - parenthesis meta x * meta y end parenthesis end line
line ell h because lemma eqTransitivity4 modus ponens ell e modus ponens ell f modus ponens ell g indeed
  meta x * (-1) * meta y = - parenthesis meta x * meta y end parenthesis end line

line ell i because lemma equality modus ponens ell h modus ponens ell c indeed
  - parenthesis meta x * meta y end parenthesis = meta x * parenthesis (-1) * meta y end parenthesis end line

line ell j because lemma eqTransitivity modus ponens ell i modus ponens ell b indeed
  - parenthesis meta x * meta y end parenthesis = meta x * - meta y end line

because lemma eqSymmetry modus ponens ell j indeed 
  meta x * - meta y = - parenthesis meta x * meta y end parenthesis qed
end math ]"


"[  math in theory system Q lemma lemma splitNumericalProduct(++) says
for all terms meta x comma meta y indeed
0 <= meta x infer 0 <= meta y infer | meta x * meta y | = | meta x | * | meta y | 
end lemma end math ]"

"[ math system Q proof of lemma splitNumericalProduct(++) reads
any term meta x comma meta y end line
line ell a premise 0 <= meta x end line
line ell b premise 0 <= meta y end line
line ell c because lemma nonnegativeFactors modus ponens ell a modus ponens ell b indeed
  0 <= meta x * meta y end line
line ell d because lemma nonnegativeNumerical modus ponens ell c indeed
  | meta x * meta y | = meta x * meta y end line

line ell e because lemma nonnegativeNumerical modus ponens ell a indeed | meta x | = meta x end line
line ell f because lemma nonnegativeNumerical modus ponens ell b indeed | meta y | = meta y end line
line ell g because lemma multiplyEquations modus ponens ell e modus ponens ell f indeed
  | meta x | * | meta y | = meta x * meta y end line
line ell h because lemma eqSymmetry modus ponens ell g indeed 
  meta x * meta y = | meta x | * | meta y | end line

because lemma eqTransitivity modus ponens ell d modus ponens ell h indeed 
  | meta x * meta y | =  | meta x | * | meta y | qed
end math ]"


"[  math in theory system Q lemma lemma splitNumericalProduct(+-) says
for all terms meta x comma meta y indeed
0 <= meta x infer meta y <= 0 infer | meta x * meta y | = | meta x | * | meta y |
end lemma end math ]"

"[ math system Q proof of lemma splitNumericalProduct(+-) reads
any term meta x comma meta y end line
line ell a premise 0 <= meta x end line
line ell b premise meta y <= 0 end line

line ell c because lemma signNumerical indeed 
 | meta x * meta y | = | - parenthesis meta x * meta y end parenthesis |  end line
line ell big c because lemma eqSymmetry modus ponens ell c indeed
 | - parenthesis meta x * meta y end parenthesis | = | meta x * meta y | end line
line ell d because lemma plusTimesMinus indeed 
  meta x * - meta y = - parenthesis meta x * meta y end parenthesis end line
line ell e because lemma sameNumerical modus ponens ell d indeed
  | meta x * - meta y | = | - parenthesis meta x * meta y end parenthesis | end line
line ell f because lemma eqTransitivity modus ponens ell e modus ponens ell big c indeed
   | meta x * - meta y | = | meta x * meta y | end line

line ell g because lemma signNumerical indeed | meta y | = | - meta y | end line
line ell big g because lemma eqSymmetry modus ponens ell g indeed | - meta y | = | meta y | end line
line ell h because lemma eqMultiplicationLeft modus ponens ell big g indeed 
  | meta x | * | - meta y | = | meta x | * | meta y | end line

line ell i because lemma nonpositiveNegated modus ponens ell b indeed 0 <= - meta y end line
line ell j because lemma splitNumericalProduct(++) modus ponens ell a modus ponens ell i indeed
  | meta x * - meta y | = | meta x | * | - meta y | end line
line ell k because lemma eqTransitivity modus ponens ell j modus ponens ell h indeed 
  | meta x * - meta y | = | meta x | * | meta y | end line

because lemma equality modus ponens ell f modus ponens ell k indeed 
  | meta x * meta y | = | meta x | * | meta y | qed
end math ]"


"[  math in theory system Q lemma lemma splitNumericalProduct says
for all terms meta x comma meta y indeed
| meta x * meta y | = | meta x | * | meta y | 
end lemma end math ]"


"[ math system Q proof of lemma splitNumericalProduct reads

block any term meta x comma meta y end line
line ell a premise 0 <= meta x end line
line ell b premise 0 <= meta y end line
because lemma splitNumericalProduct(++) modus ponens ell a modus ponens ell b indeed
  | meta x * meta y | = | meta x | * | meta y | end line
line ell big a end block

block any term meta x comma meta y end line
line ell a premise 0 <= meta x end line
line ell b premise meta y <= 0 end line
because lemma splitNumericalProduct(+-) modus ponens ell a modus ponens ell b indeed
  | meta x * meta y | = | meta x | * | meta y | end line
line ell big b end block

block any term meta x comma meta y end line
line ell a premise meta x <= 0 end line
line ell b premise 0 <= meta y end line
line ell c because lemma splitNumericalProduct(+-) modus ponens ell b modus ponens ell a indeed
  | meta y * meta x | = | meta y | * | meta x | end line

line ell d because axiom timesCommutativity indeed meta x * meta y = meta y * meta x end line
line ell e because lemma sameNumerical modus ponens ell d indeed
  | meta x * meta y | = | meta y * meta x | end line

line ell f because axiom timesCommutativity indeed 
  | meta y | * | meta x | = | meta x | * | meta y |  end line

because lemma eqTransitivity4 modus ponens ell e modus ponens ell c modus ponens ell f indeed
  | meta x * meta y | = | meta x | * | meta y | end line
line ell big c end block

block any term meta x comma meta y end line
line ell a premise meta x <= 0 end line
line ell b premise meta y <= 0 end line
line ell c because lemma nonpositiveNegated modus ponens ell a indeed 0 <= - meta x end line
line ell d because lemma nonpositiveNegated modus ponens ell b indeed 0 <= - meta y end line
line ell e because lemma splitNumericalProduct(++) modus ponens ell c modus ponens ell d indeed
  | - meta x * - meta y | = | - meta x | * | - meta y | end line

line ell f because lemma minusTimesMinus indeed - meta x * - meta y = meta x * meta y end line
line ell g because lemma sameNumerical modus ponens ell f indeed 
  | - meta x * - meta y | = | meta x * meta y | end line
line ell h because lemma eqSymmetry modus ponens ell g indeed 
  | meta x * meta y | = | - meta x * - meta y | end line

line ell i because lemma signNumerical indeed | meta x | = | - meta x | end line
line ell j because lemma signNumerical indeed | meta y | = | - meta y | end line
line ell k because lemma multiplyEquations modus ponens ell i modus ponens ell j indeed
  | meta x | * | meta y | = | - meta x | * | - meta y | end line
line ell l because lemma eqSymmetry modus ponens ell k indeed
  | - meta x | * | - meta y | = | meta x | * | meta y | end line
because lemma eqTransitivity4 modus ponens ell h modus ponens ell e modus ponens ell l indeed
  | meta x * meta y | = | meta x | * | meta y | end line
line ell big d end block

any term meta x comma meta y end line
line ell a because 1rule deduction modus ponens ell big a indeed
  0 <= meta x imply 0 <= meta y imply | meta x * meta y | = | meta x | * | meta y | end line
line ell b because 1rule deduction modus ponens ell big b indeed
  0 <= meta x imply meta y <= 0 imply | meta x * meta y | = | meta x | * | meta y | end line
line ell c because 1rule deduction modus ponens ell big c indeed
  meta x <= 0 imply 0 <= meta y imply | meta x * meta y | = | meta x | * | meta y | end line
line ell d because 1rule deduction modus ponens ell big d indeed
  meta x <= 0 imply meta y <= 0 imply | meta x * meta y | = | meta x | * | meta y | end line

line ell e because lemma from leqGeq modus ponens ell a modus ponens ell c indeed
  0 <= meta y imply | meta x * meta y | = | meta x | * | meta y | end line
line ell f because lemma from leqGeq modus ponens ell b modus ponens ell d indeed
  meta y <= 0 imply | meta x * meta y | = | meta x | * | meta y | end line

because lemma from leqGeq modus ponens ell e modus ponens ell f indeed  
| meta x * meta y | = | meta x | * | meta y | qed
end math ]"

"[  math in theory system Q lemma lemma three2threeFactors says
for all terms meta x comma meta y comma meta z indeed
meta x * meta y * meta z = meta x * meta z * meta y
end lemma end math ]"

"[ math system Q proof of lemma three2threeFactors reads
any term meta x comma meta y comma meta z end line
line ell a because axiom timesCommutativity indeed meta y * meta z = meta z * meta y end line
line ell b because lemma three2twoFactors modus ponens ell a indeed
  meta x * meta y * meta z = meta x * parenthesis meta z * meta y end parenthesis end line

line ell c because axiom timesAssociativity indeed
  meta x * meta z * meta y = meta x * parenthesis meta z * meta y end parenthesis end line 
line ell d because lemma eqSymmetry modus ponens ell c indeed
  meta x * parenthesis meta z * meta y end parenthesis = meta x * meta z * meta y end line

because lemma eqTransitivity modus ponens ell b modus ponens ell d indeed  
  meta x * meta y * meta z = meta x * meta z * meta y qed
end math ]"


"[  math in theory system Q lemma lemma distributionOutLeft says
for all terms meta x comma meta y comma meta z indeed
meta y * meta x + meta z * meta x = meta x * parenthesis meta y + meta z end parenthesis
end lemma end math ]"

"[ math system Q proof of lemma distributionOutLeft reads
any term meta x comma meta y comma meta z end line
line ell a because axiom timesCommutativity indeed meta y * meta x = meta x * meta y end line
line ell b because axiom timesCommutativity indeed meta z * meta x = meta x * meta z end line
line ell c because lemma addEquations modus ponens ell a modus ponens ell b indeed
  meta y * meta x + meta z * meta x = meta x * meta y + meta x * meta z end line
line ell d because lemma distributionOut indeed
  meta x * meta y + meta x * meta z = meta x * parenthesis meta y + meta z end parenthesis end line

because lemma eqTransitivity modus ponens ell c modus ponens ell d indeed 
  meta y * meta x + meta z * meta x = meta x * parenthesis meta y + meta z end parenthesis qed
end math ]"



"[  math in theory system Q lemma lemma nonzeroFactors says
for all terms meta x comma meta y indeed

meta x != 0 infer meta y != 0 infer meta x * meta y != 0 
end lemma end math ]"

"[ math system Q proof of lemma nonzeroFactors reads
any term meta x comma meta y end line
line ell a premise meta x != 0 end line
line ell b premise meta y != 0 end line
line ell c because lemma neqMultiplication modus ponens ell b modus ponens ell a indeed
  meta x * meta y != 0 * meta y end line

line ell d because axiom timesCommutativity indeed 0 * meta y = meta y * 0 end line
line ell e because lemma x*0=0 indeed meta y * 0 = 0 end line ""; XX muligt lemma her
line ell f because lemma eqTransitivity modus ponens ell d modus ponens ell e indeed 0 * meta y = 0 end line

because lemma subNeqRight modus ponens ell f modus ponens ell c indeed 
 meta x * meta y != 0 qed
end math ]"

"[  math in theory system Q lemma lemma positiveFactors says
for all terms meta x comma meta y indeed
0 < meta x infer 0 < meta y infer 0 < meta x * meta y
end lemma end math ]"

"[ math system Q proof of lemma positiveFactors reads
any term meta x comma meta y end line
line ell a premise 0 < meta x end line
line ell b premise 0 < meta y end line

line ell c because 1rule repetition modus ponens ell a indeed 
  0 <= meta x and0 0 != meta x end line ""; XXREP <
line ell d because prop lemma first conjunct modus ponens ell c indeed 0 <= meta x end line
line ell e because prop lemma second conjunct modus ponens ell c indeed 0 != meta x end line
line ell f because lemma neqSymmetry modus ponens ell e indeed meta x != 0 end line

line ell g because 1rule repetition modus ponens ell b indeed 
  0 <= meta y and0 0 != meta y end line ""; XXREP <
line ell h because prop lemma first conjunct modus ponens ell g indeed 0 <= meta y end line
line ell i because prop lemma second conjunct modus ponens ell g indeed 0 != meta y end line
line ell j because lemma neqSymmetry modus ponens ell i indeed meta y != 0 end line

line ell k because lemma nonnegativeFactors modus ponens ell d modus ponens ell h indeed
  0 <= meta x * meta y end line
line ell l because lemma nonzeroFactors modus ponens ell f modus ponens ell j indeed
  meta x * meta y != 0 end line
line ell m because lemma neqSymmetry modus ponens ell l indeed 0 != meta x * meta y end line

line ell n because prop lemma join conjuncts modus ponens ell k modus ponens ell m indeed
  0 <= meta x * meta y and0 0 != meta x * meta y end line
because 1rule repetition modus ponens ell n indeed 0 < meta x * meta y qed ""; XXREP <
end math ]"



"[ math in theory system Q lemma lemma seriesSubsetCP says
for all terms meta fx comma meta sy indeed
isSeries( meta fx , meta sy ) infer
isSubset( meta fx , cartProd( N , meta sy ) )
end lemma end math ]"

"[ math system Q proof of lemma seriesSubsetCP reads
block any term meta fx comma meta sy end line
line ell a premise isSeries( meta fx , meta sy ) end line
line ell b premise object s1 in0 meta fx end line
line ell c because lemma fromSeries modus ponens ell a indeed
  ( for all object r1 indeed ( object r1 in0 meta fx imply exist0 object op1 indeed exist0 object op2 indeed 
    object op1 in0 N and0 object op2 in0 meta sy and0 object r1 = (o object op1 , object op2 ) ) ) and0
  ( for all object f1 comma object f2 comma object f3 comma object f4 indeed
    ( (o object f1 , object f2 ) in0 meta fx imply (o object f3 , object f4 ) in0 meta fx imply
    object f1 = object f3 imply object f2 = object f4 ) ) and0
  for all object s1 indeed 
    ( object s1 in0 N imply exist0 object s2 indeed (o object s1 , object s2 ) in0 meta fx ) end line
line ell d because prop lemma first conjunct modus ponens ell c indeed ""; XX problem her
  ( for all object r1 indeed ( object r1 in0 meta fx imply exist0 object op1 indeed exist0 object op2 indeed 
    object op1 in0 N and0 object op2 in0 meta sy and0 object r1 = (o object op1 , object op2 ) ) ) and0
  for all object f1 comma object f2 comma object f3 comma object f4 indeed
    ( (o object f1 , object f2 ) in0 meta fx imply (o object f3 , object f4 ) in0 meta fx imply
    object f1 = object f3 imply object f2 = object f4 ) end line
line ell e because prop lemma first conjunct modus ponens ell d indeed
  for all object r1 indeed ( object r1 in0 meta fx imply exist0 object op1 indeed exist0 object op2 indeed 
    object op1 in0 N and0 object op2 in0 meta sy and0 object r1 = (o object op1 , object op2 ) ) end line
line ell f because lemma a4 at object s1 modus ponens ell e indeed 
  object s1 in0 meta fx imply exist0 object op1 indeed exist0 object op2 indeed 
  object op1 in0 N and0 object op2 in0 meta sy and0 object s1 = (o object op1 , object op2 ) end line
line ell g because 1rule mp modus ponens ell f modus ponens ell b indeed
  exist0 object op1 indeed exist0 object op2 indeed 
  object op1 in0 N and0 object op2 in0 meta sy and0 object s1 = (o object op1 , object op2 ) end line

block any term meta sy end line
line ell a premise 
  object op1 in0 N and0 object op2 in0 meta sy and0 object s1 = (o object op1 , object op2 ) end line
line ell b because prop lemma first conjunct modus ponens ell a indeed
  object op1 in0 N and0 object op2 in0 meta sy end line
line ell c because prop lemma first conjunct modus ponens ell b indeed  object op1 in0 N end line
line ell d because prop lemma second conjunct modus ponens ell b indeed
  object op2 in0 meta sy end line
line ell e because prop lemma second conjunct modus ponens ell a indeed
  object s1 = (o object op1 , object op2 ) end line
line ell f because lemma eqSymmetry modus ponens ell e indeed
  (o object op1 , object op2 ) = object s1 end line

line ell g because lemma toCartProd modus ponens ell c modus ponens ell d indeed
  (o object op1 , object op2 ) in0 cartProd( N , meta sy ) end line
because lemma sameMember modus ponens ell f modus ponens ell g indeed
  object s1 in0 cartProd( N , meta sy ) end line
line ell big x end block

line ell h because 1rule deduction modus ponens ell big x indeed
  object op1 in0 N and0 object op2 in0 meta sy and0 object s1 = (o object op1 , object op2 ) imply
  object s1 in0 cartProd( N , meta sy ) end line
because pred lemma 2exist mp modus ponens ell h modus ponens ell g indeed
  object s1 in0 cartProd( N , meta sy ) end line
line ell big a end block

any term meta fx comma meta sy end line

line ell a because 1rule deduction modus ponens ell big a indeed
  isSeries( meta fx , meta sy ) imply for all object s1 indeed (
  object s1 in0 meta fx imply object s1 in0 cartProd( N , meta sy ) ) end line

line ell c premise isSeries( meta fx , meta sy ) end line

line ell d because 1rule mp modus ponens ell a modus ponens ell c indeed
  for all object s1 indeed ( object s1 in0 meta fx imply object s1 in0 cartProd( N , meta sy ) ) end line
because 1rule repetition modus ponens ell d indeed 
  isSubset( meta fx , cartProd( N , meta sy ) ) qed ""; XXREP isSubset
end math ]"


"[ math in theory system Q lemma lemma fromCartProd says
for all terms meta sx comma meta sx1 comma meta sy comma meta sy1 indeed
(o meta sx , meta sy ) in0 cartProd( meta sx1 , meta sy1 ) infer
meta sx in0 meta sx1 and0 meta sy in0 meta sy1
end lemma end math ]"

"[ math system Q proof of lemma fromCartProd reads
any term meta sx comma meta sx1 comma meta sy comma meta sy1 end line
line ell a premise (o meta sx , meta sy ) in0 cartProd( meta sx1 , meta sy1 ) end line
line ell b because 1rule repetition modus ponens ell a indeed (o meta sx , meta sy ) in0 ""; XXREP cartProd
  the set of ph in P( P( binaryUnion( meta  sx1 , meta sy1 ) ) ) such that 
  isOrderedPair( ph1 , meta sx1 , meta sy1 ) end set end line
line ell c because lemma separation2formula modus ponens ell b indeed
  (o meta sx , meta sy ) in0 P( P( binaryUnion( meta  sx1 , meta sy1 ) ) ) and0
  isOrderedPair( (o meta sx , meta sy ) , meta sx1 , meta sy1 ) end line
line ell d because prop lemma second conjunct modus ponens ell c indeed
  isOrderedPair( (o meta sx , meta sy ) , meta sx1 , meta sy1 ) end line
line ell e because 1rule repetition modus ponens ell d indeed ""; XXREP isOrderedPair
  exist0 object op1 indeed exist0 object op2 indeed 
  object op1 in0 meta sx1 and0 object op2 in0 meta sy1 and0 
  (o meta sx , meta sy ) = (o object op1 , object op2 ) end line

block any term meta sx comma meta sx1 comma meta sy comma meta sy1 end line
line ell a premise object op1 in0 meta sx1 and0 object op2 in0 meta sy1 and0 
  (o meta sx , meta sy ) = (o object op1 , object op2 ) end line
line ell b because prop lemma first conjunct modus ponens ell a indeed
  object op1 in0 meta sx1 and0 object op2 in0 meta sy1 end line
line ell c because prop lemma first conjunct modus ponens ell b indeed object op1 in0 meta sx1 end line
line ell d because prop lemma second conjunct modus ponens ell b indeed object op2 in0 meta sy1 end line
line ell e because prop lemma second conjunct modus ponens ell a indeed 
  (o meta sx , meta sy ) = (o object op1 , object op2 ) end line

line ell f because lemma fromOrderedPair(1) modus ponens ell e indeed meta sx = object op1 end line
line ell g because lemma sameMember(2) modus ponens ell f modus ponens ell c indeed
  meta sx in0 meta sx1 end line

line ell h because lemma fromOrderedPair(2) modus ponens ell e indeed meta sy = object op2 end line
line ell i because lemma sameMember(2) modus ponens ell h modus ponens ell d indeed
  meta sy in0 meta sy1 end line
because prop lemma join conjuncts modus ponens ell g modus ponens ell i indeed 
  meta sx in0 meta sx1 and0 meta sy in0 meta sy1 end line
line ell big a end block

line ell f because 1rule deduction modus ponens ell big a indeed
  object op1 in0 meta sx1 and0 object op2 in0 meta sy1 and0 
  (o meta sx , meta sy ) = (o object op1 , object op2 ) imply
  meta sx in0 meta sx1 and0 meta sy in0 meta sy1 end line

because pred lemma 2exist mp modus ponens ell f modus ponens ell e indeed 
  meta sx in0 meta sx1 and0 meta sy in0 meta sy1 qed
end math ]"
 


"[ math in theory system Q lemma lemma fromCartProd(1) says
for all terms meta sx comma meta sx1 comma meta sy comma meta sy1 indeed
(o meta sx , meta sy ) in0 cartProd( meta sx1 , meta sy1 ) infer
meta sx in0 meta sx1
end lemma end math ]"

"[ math system Q proof of lemma fromCartProd(1) reads
any term meta sx comma meta sx1 comma meta sy comma meta sy1 end line
line ell a premise (o meta sx , meta sy ) in0 cartProd( meta sx1 , meta sy1 ) end line
line ell b because lemma fromCartProd modus ponens ell a indeed
  meta sx in0 meta sx1 and0 meta sy in0 meta sy1 end line
because prop lemma first conjunct modus ponens ell b indeed meta sx in0 meta sx1 qed
end math ]"

"[ math in theory system Q lemma lemma fromCartProd(2) says
for all terms meta sx comma meta sx1 comma meta sy comma meta sy1 indeed
(o meta sx , meta sy ) in0 cartProd( meta sx1 , meta sy1 ) infer
meta sy in0 meta sy1
end lemma end math ]"

"[ math system Q proof of lemma fromCartProd(2) reads
any term meta sx comma meta sx1 comma meta sy comma meta sy1 end line
line ell a premise (o meta sx , meta sy ) in0 cartProd( meta sx1 , meta sy1 ) end line
line ell b because lemma fromCartProd modus ponens ell a indeed
  meta sx in0 meta sx1 and0 meta sy in0 meta sy1 end line
because prop lemma second conjunct modus ponens ell b indeed meta sy in0 meta sy1 qed
end math ]"



"[ math in theory system Q lemma lemma valueType says
for all terms meta m comma meta fx comma meta sy indeed
meta m in0 N infer
isSeries( meta fx , meta sy ) infer
[ meta fx ; meta m ] in0 meta sy
end lemma end math ]"

"[ math system Q proof of lemma valueType reads
any term meta m comma meta fx comma meta sy end line
line ell a premise meta m in0 N end line
line ell b premise isSeries( meta fx , meta sy ) end line
line ell c because lemma memberOfSeries modus ponens ell a modus ponens ell b indeed
  (o meta m , [ meta fx ; meta m ] ) in0 meta fx end line

line ell d because lemma seriesSubsetCP modus ponens ell b indeed
  isSubset( meta fx , cartProd( N , meta sy ) ) end line
line ell e because lemma fromSubset modus ponens ell d modus ponens ell c indeed
  (o meta m , [ meta fx ; meta m ] ) in0 cartProd( N , meta sy ) end line
because lemma fromCartProd(2) modus ponens ell e indeed
  [ meta fx ; meta m ] in0 meta sy  qed
end math ]"

"[ math in theory system Q lemma lemma productIsFunction says
for all terms meta m1 comma meta m2 comma meta fx comma meta fy indeed ""; XX ekstra variable
for all object f1 comma object f2 comma object f3 comma object f4 indeed
  ( (o object f1 , object f2 ) in0 meta fx *f meta fy imply 
  (o object f3 , object f4 ) in0 meta fx *f meta fy imply 
  object f1 = object f3 imply object f2 = object f4 )
end lemma end math ]"

"[ math system Q proof of lemma productIsFunction reads
block any term meta m1 comma meta m2 comma meta fx comma meta fy end line
line ell a premise object f1 = object f3 end line
line ell b premise (o object f1 , object f2 ) =
  (o meta m1 , [ meta fx ; meta m1 ] * [ meta fy ; meta m1 ] ) end line
line ell c premise (o object f3 , object f4 ) =
  (o meta m2 , [ meta fx ; meta m2 ] * [ meta fy ; meta m2 ] ) end line
line ell d because lemma fromOrderedPair(1) modus ponens ell b indeed
  object f1 = meta m1 end line
line ell e because lemma eqSymmetry modus ponens ell d indeed meta m1 = object f1 end line
line ell f because lemma fromOrderedPair(1) modus ponens ell c indeed
  object f3 = meta m2 end line
line ell g because lemma eqTransitivity4 modus ponens ell e modus ponens ell a modus ponens ell f indeed
  meta m1 = meta m2 end line

line ell h because lemma sameSeries modus ponens ell g indeed
  [ meta fx ; meta m1 ] = [ meta fx ; meta m2 ] end line
line ell i because lemma sameSeries modus ponens ell g indeed
  [ meta fy ; meta m1 ] = [ meta fy ; meta m2 ] end line
line ell j because lemma multiplyEquations modus ponens ell h modus ponens ell i indeed
  [ meta fx ; meta m1 ] * [ meta fy ; meta m1 ] =
  [ meta fx ; meta m2 ] * [ meta fy ; meta m2 ] end line

line ell k because lemma fromOrderedPair(2) modus ponens ell b indeed
  object f2 = [ meta fx ; meta m1 ] * [ meta fy ; meta m1 ] end line

line ell l because lemma fromOrderedPair(2) modus ponens ell c indeed
  object f4 = [ meta fx ; meta m2 ] * [ meta fy ; meta m2 ] end line
line ell m because lemma eqSymmetry modus ponens ell l indeed
  [ meta fx ; meta m2 ] * [ meta fy ; meta m2 ] = object f4 end line
because lemma eqTransitivity4 modus ponens ell k modus ponens ell j modus ponens ell m indeed
  object f2 = object f4 end line
line ell big a end block


any term meta m1 comma meta m2 comma meta fx comma meta fy end line

because 1rule deduction modus ponens ell big a indeed 
for all object f1 comma object f2 comma object f3 comma object f4 indeed
  ( (o object f1 , object f2 ) in0 meta fx *f meta fy imply 
  (o object f3 , object f4 ) in0 meta fx *f meta fy imply 
  object f1 = object f3 imply object f2 = object f4 ) qed
end math ]"

"[ math in theory system Q lemma lemma productIsTotal says ""; XX ekstra variable
  for all terms meta m comma meta fx comma meta fy indeed
  for all object s1 indeed ( object s1 in0 N imply 
  exist0 object s2 indeed (o object s1 , object s2 ) in0 meta fx *f meta fy ) 
end lemma end math ]"

"[ math system Q proof of lemma productIsTotal reads
block any term meta m comma meta fx comma meta fy end line
line ell a premise object s1 in0 N end line
line ell b because axiom seriesType indeed isSeries( meta fx , Q ) end line ""; XX stinker lidt...
line ell x because axiom seriesType indeed isSeries( meta fy , Q ) end line
line ell c because lemma valueType modus ponens ell a modus ponens ell b indeed 
  [ meta fx ; object s1 ] in0 Q end line
line ell d because lemma valueType modus ponens ell a modus ponens ell x indeed
  [ meta fy ; object s1 ] in0 Q end line

line ell e because 1rule Qclosed(Multiplication) modus ponens ell c modus ponens ell d indeed
  [ meta fx ; object s1 ] * [ meta fy ; object s1 ] in0 Q end line
line ell f because lemma toCartProd modus ponens ell a modus ponens ell e indeed
  (o object s1 , [ meta fx ; object s1 ] * [ meta fy ; object s1 ] ) in0 cartProd( N , Q ) end line

line ell g because lemma eqReflexivity indeed
  (o object s1 , [ meta fx ; object s1 ] * [ meta fy ; object s1 ] ) =
  (o object s1 , [ meta fx ; object s1 ] * [ meta fy ; object s1 ] ) end line
line ell h because pred lemma intro exist at object s1 modus ponens ell g indeed
  exist0 meta m indeed 
  (o object s1 , [ meta fx ; object s1 ] * [ meta fy ; object s1 ] ) =
  (o meta m , [ meta fx ; meta m ] * [ meta fy ; meta m ] ) end line

line ell i because lemma formula2separation modus ponens ell f modus ponens ell h indeed
  (o object s1 , [ meta fx ; object s1 ] * [ meta fy ; object s1 ] ) in0
   the set of ph in cartProd( N , Q ) such that exist0 meta m indeed 
     ph5 = (o meta m , [ meta fx ; meta m ] * [ meta fy ; meta m ] ) end set end line
line ell j because 1rule repetition modus ponens ell i indeed ""; XXREP *f
  (o object s1 , [ meta fx ; object s1 ] * [ meta fy ; object s1 ] ) in0 meta fx *f meta fy end line
 because pred lemma intro exist at [ meta fx ; object s1 ] * [ meta fy ; object s1 ] 
   modus ponens ell j indeed exist0 object s2 indeed    
  (o object s1 , object s2 ) in0 meta fx *f meta fy end line
line ell big a end block

any term meta m comma meta fx comma meta fy end line

because 1rule deduction modus ponens ell big a indeed 
  for all object s1 indeed ( object s1 in0 N imply 
  exist0 object s2 indeed (o object s1 , object s2 ) in0 meta fx *f meta fy ) qed
end math ]"


"[ math in theory system Q lemma lemma productIsRationalSeries says
for all terms meta m comma meta m1 comma meta m2 comma meta fx comma meta fy indeed ""; XX ekstra variable
isSeries( meta fx *f meta fy , Q )
end lemma end math ]"

"[ math system Q proof of lemma productIsRationalSeries reads
any term meta m comma meta m1 comma meta m2 comma meta fx comma meta fy end line

line ell a because lemma CPseparationIsRelation indeed 
  isRelation( meta fx *f meta fy , N , Q ) end line

line ell b because lemma productIsFunction indeed 
for all object f1 comma object f2 comma object f3 comma object f4 indeed
  ( (o object f1 , object f2 ) in0 meta fx *f meta fy imply 
  (o object f3 , object f4 ) in0 meta fx *f meta fy imply 
  object f1 = object f3 imply object f2 = object f4 ) end line

line ell c because lemma productIsTotal indeed 
  for all object s1 indeed ( object s1 in0 N imply 
  exist0 object s2 indeed (o object s1 , object s2 ) in0 meta fx *f meta fy ) end line

because lemma toSeries modus ponens ell a modus ponens ell b modus ponens ell c indeed 
  isSeries( meta fx *f meta fy , Q ) qed
end math ]"


"[  math in theory system Q lemma lemma timesF says
for all terms meta m comma meta fx comma meta fy indeed
typeNat( meta m ) endorse
typeSeries( meta fx , Q ) endorse
typeSeries( meta fy , Q ) endorse
[ meta fx *f meta fy ; meta m ] = [ meta fx ; meta m ] * [ meta fy ; meta m ]
end lemma end math ]"

"[ math system Q proof of lemma timesF reads
any term meta m comma meta fx comma meta fy end line
line ell a side condition typeNat( meta m ) end  line
line ell b side condition typeSeries( meta fx , Q ) end line
line ell c side condition typeSeries( meta fy , Q ) end line
line ell d because axiom natType indeed meta m in0 N end line
line ell x because axiom seriesType indeed isSeries( meta fx , Q ) end line
line ell y because axiom seriesType indeed isSeries( meta fy , Q ) end line

line ell e because lemma productIsRationalSeries indeed isSeries( meta fx *f meta fy , Q ) end line
line ell f because lemma memberOfSeries modus ponens ell d modus ponens ell e indeed 
  (o meta m , [ meta fx *f meta fy ; meta m ] ) in0 meta fx *f meta fy end line

line ell g because lemma valueType modus ponens ell d modus ponens ell x indeed 
  [ meta fx ; meta m ] in0 Q end line
line ell h because lemma valueType modus ponens ell d modus ponens ell y indeed 
  [ meta fy ; meta m ] in0 Q end line
line ell i because 1rule Qclosed(Multiplication) modus ponens ell g modus ponens ell h indeed
  [ meta fx ; meta m ] * [ meta fy ; meta m ] in0 Q end line

line ell j because lemma toCartProd modus ponens ell d modus ponens ell i indeed
  (o meta m , [ meta fx ; meta m ] * [ meta fy ; meta m ] ) in0 cartProd( N , Q ) end line
line ell k because lemma eqReflexivity indeed 
  (o meta m , [ meta fx ; meta m ] * [ meta fy ; meta m ] ) =
  (o meta m , [ meta fx ; meta m ] * [ meta fy ; meta m ] ) end line
line ell l because pred lemma intro exist at meta m modus ponens ell k indeed
  exist0 meta m indeed (o meta m , [ meta fx ; meta m ] * [ meta fy ; meta m ] ) =
  (o meta m , [ meta fx ; meta m ] * [ meta fy ; meta m ] ) end line
line ell m because lemma formula2separation modus ponens ell j modus ponens ell l indeed
  (o meta m , [ meta fx ; meta m ] * [ meta fy ; meta m ] ) in0 meta fx *f meta fy end line

line ell n because lemma eqReflexivity indeed meta m = meta m end line
because lemma uniqueMember modus ponens ell e modus ponens ell f modus ponens ell m modus ponens ell n
  indeed [ meta fx *f meta fy ; meta m ] = [ meta fx ; meta m ] * [ meta fy ; meta m ] qed
end math ]"



"[ math in theory system Q lemma lemma sameOrderedPair says
for all terms meta sx comma meta sx1 comma meta sy comma meta sy1 indeed
meta sx = meta sx1 infer meta sy = meta sy1 infer (o meta sx , meta sy ) = (o meta sx1 , meta sy1 )
end lemma end math ]"

"[ math system Q proof of lemma sameOrderedPair reads
any term meta sx comma meta sx1 comma meta sy comma meta sy1 end line
line ell a premise meta sx = meta sx1 end line 
line ell b premise meta sy = meta sy1 end line

line ell c because lemma same singleton modus ponens ell a indeed (s meta sx ) = (s meta sx1 ) end line
line ell d because lemma same pair modus ponens ell a modus ponens ell b indeed
  (p meta sx , meta sy ) = (p meta sx1 , meta sy1 ) end line
line ell e because lemma same pair modus ponens ell c modus ponens ell d indeed
  (p (s meta sx ) , (p meta sx , meta sy ) ) = (p (s meta sx1 ) , (p meta sx1 , meta sy1 ) ) end line
because 1rule repetition modus ponens ell e indeed 
  (o meta sx , meta sy ) = (o meta sx1 , meta sy1 ) qed ""; XXREP (o )
end math ]"

"[ math in theory system Q lemma lemma inSeries helper says
for all terms meta m comma meta fx comma meta sx comma meta sy indeed
  meta sy in0 meta fx imply
  isSeries( meta fx , meta sx ) imply
  object op1 in0 N and0 object op2 in0 meta sx and0 meta sy = (o object op1 , object op2 ) imply
  exist0 meta m indeed meta sy = (o meta m , [ meta fx ; meta m ] ) 
end lemma end math ]"

"[ math system Q proof of lemma inSeries helper reads
block any term meta m comma meta fx comma meta sx comma meta sy end line
line ell e premise meta sy in0 meta fx end line
line ell a premise isSeries( meta fx , meta sx ) end line
line ell k premise 
  object op1 in0 N and0 object op2 in0 meta sx and0 meta sy = (o object op1 , object op2 ) end line
line ell b because prop lemma first conjunct modus ponens ell k indeed 
  object op1 in0 N and0 object op2 in0 meta sx end line
line ell c because prop lemma first conjunct modus ponens ell b indeed 
  object op1 in0 N end line
line ell d because prop lemma second conjunct modus ponens ell k indeed
  meta sy = (o object op1 , object op2 ) end line
line ell f because lemma sameMember modus ponens ell d modus ponens ell e indeed
  (o object op1 , object op2 ) in0 meta fx end line


line ell l because lemma memberOfSeries modus ponens ell c modus ponens ell a indeed
  (o object op1 , [ meta fx ; object op1 ] ) in0 meta fx end line
line ell g because lemma eqReflexivity indeed object op1 = object op1 end line
line ell n because lemma uniqueMember modus ponens ell a modus ponens ell f modus ponens ell l
  modus ponens ell g indeed object op2 = [ meta fx ; object op1 ] end line

line ell o because lemma sameOrderedPair modus ponens ell g modus ponens ell n indeed
  (o object op1 , object op2 ) = (o object op1 , [ meta fx ; object op1 ] ) end line
line ell p because lemma eqTransitivity modus ponens ell d modus ponens ell o indeed
  meta sy = (o object op1 , [ meta fx ; object op1 ] ) end line
because pred lemma intro exist at object op1 modus ponens ell p indeed
  exist0 meta m indeed meta sy = (o meta m , [ meta fx ; meta m ] ) end line
line ell big a end block

any term meta m comma meta fx comma meta sx comma meta sy end line

because 1rule deduction modus ponens ell big a indeed
  meta sy in0 meta fx imply
  isSeries( meta fx , meta sx ) imply
  object op1 in0 N and0 object op2 in0 meta sx and0 meta sy = (o object op1 , object op2 ) imply
  exist0 meta m indeed meta sy = (o meta m , [ meta fx ; meta m ] ) qed
end math ]"

"[ math in theory system Q lemma lemma inSeries says
for all terms meta m comma meta fx comma meta sx comma meta sy indeed
typeSeries( meta fx , meta sx ) endorse
meta sy in0 meta fx infer
exist0 meta m indeed meta sy = (o meta m , [ meta fx ; meta m ] )
end lemma end math ]"

"[ math system Q proof of lemma inSeries reads
any term meta m comma meta fx comma meta sx comma meta sy end line
line ell a side condition typeSeries( meta fx , meta sx ) end line
line ell b premise meta sy in0 meta fx end line
line ell c because axiom seriesType modus probans ell a indeed isSeries( meta fx , meta sx ) end line
line ell d because 1rule repetition modus ponens ell c indeed ""; XXREP isSeries
  isFunction( meta fx , N , meta sx ) and0
  for all object s1 indeed 
  ( object s1 in0 N imply exist0 object s2 indeed (o object s1 , object s2 ) in0 meta fx ) end line
line ell e because prop lemma first conjunct modus ponens ell d indeed
  isFunction( meta fx , N , meta sx ) end line
line ell f because 1rule repetition modus ponens ell e indeed ""; XXREP isFunction
  isRelation( meta fx , N , meta sx ) and0
  for all object f1 comma object f2 comma object f3 comma object f4 indeed
  ( (o object f1 , object f2 ) in0 meta fx imply (o object f3 , object f4 ) in0 meta fx imply
  object f1 = object f3 imply object f2 = object f4 ) end line
line ell g because prop lemma first conjunct modus ponens ell f indeed
  isRelation( meta fx , N , meta sx ) end line
line ell h because 1rule repetition modus ponens ell g indeed ""; XXREP isRelation
  for all object r1 indeed ( object r1 in0 meta fx imply isOrderedPair( object r1 , N , meta sx ) ) end line
line ell i because lemma a4 at meta sy modus ponens ell h indeed
  meta sy in0 meta fx imply isOrderedPair( meta sy , N , meta sx ) end line
line ell j because 1rule mp modus ponens ell i modus ponens ell b indeed
  isOrderedPair( meta sy , N , meta sx ) end line
line ell k because 1rule repetition modus ponens ell j indeed ""; XXREP isOrderedPair
  exist0 object op1 indeed exist0 object op2 indeed ""; XX... comma
  object op1 in0 N and0 object op2 in0 meta sx and0 meta sy = (o object op1 , object op2 ) end line

line ell l because lemma inSeries helper indeed
  meta sy in0 meta fx imply
  isSeries( meta fx , meta sx ) imply
  object op1 in0 N and0 object op2 in0 meta sx and0 meta sy = (o object op1 , object op2 ) imply
  exist0 meta m indeed meta sy = (o meta m , [ meta fx ; meta m ] ) end line
line ell m because prop lemma mp2 modus ponens ell l modus ponens ell b modus ponens ell c indeed  
  object op1 in0 N and0 object op2 in0 meta sx and0 meta sy = (o object op1 , object op2 ) imply
  exist0 meta m indeed meta sy = (o meta m , [ meta fx ; meta m ] ) end line
because pred lemma 2exist mp modus ponens ell m modus ponens ell k indeed
  exist0 meta m indeed meta sy = (o meta m , [ meta fx ; meta m ] ) qed
end math ]"

"[  math in theory system Q lemma lemma to=f subset helper says
for all terms meta m comma meta fx comma meta fy indeed 
  isSeries( meta fy , Q ) imply
  for all meta m indeed [ meta fx ; meta m ] = [ meta fy ; meta m ] imply
  object s1 = (o meta m , [ meta fx ; meta m ] ) imply
  object s1 in0 meta fy
end lemma end math ]"

"[ math system Q proof of lemma to=f subset helper reads
block any term meta m comma meta fx comma meta fy end line
line ell b premise isSeries( meta fy , Q ) end line
line ell c premise for all meta m indeed [ meta fx ; meta m ] = [ meta fy ; meta m ] end line
line ell e premise  
  object s1 = (o meta m , [ meta fx ; meta m ] ) end line
line ell f because lemma eqReflexivity indeed meta m = meta m end line
line ell g because lemma a4 at meta m modus ponens ell c indeed 
  [ meta fx ; meta m ] = [ meta fy ; meta m ] end line
line ell h because lemma sameOrderedPair modus ponens ell f modus ponens ell g indeed
  (o meta m , [ meta fx ; meta m ] ) = (o meta m , [ meta fy ; meta m ] ) end line

line ell i because lemma eqTransitivity modus ponens ell e modus ponens ell h indeed
  object s1 = (o meta m , [ meta fy ; meta m ] ) end line
line ell j because lemma eqSymmetry modus ponens ell i indeed
  (o meta m , [ meta fy ; meta m ] ) = object s1 end line

line ell k because axiom natType indeed meta m in0 N end line

line ell l because lemma memberOfSeries modus ponens ell k modus ponens ell b indeed
  (o meta m , [ meta fy ; meta m ] ) in0 meta fy end line

because lemma sameMember modus ponens ell j modus ponens ell l indeed
  object s1 in0 meta fy end line
line ell big a end block

any term meta m comma meta fx comma meta fy end line

because 1rule deduction modus ponens ell big a indeed
  isSeries( meta fy , Q ) imply
  for all meta m indeed [ meta fx ; meta m ] = [ meta fy ; meta m ] imply
  object s1 = (o meta m , [ meta fx ; meta m ] ) imply
  object s1 in0 meta fy qed
end math ]"

"[  math in theory system Q lemma lemma  to=f subset says
for all terms meta m comma meta fx comma meta fy indeed 
typeSeries( meta fx , Q ) endorse
typeSeries( meta fy , Q ) endorse
for all meta m indeed [ meta fx ; meta m ] = [ meta fy ; meta m ] infer
isSubset( meta fx , meta fy )
end lemma end math ]"

"[ math system Q proof of lemma to=f subset reads
block any term meta m comma meta fx comma meta fy end line
line ell a premise isSeries( meta fx , Q ) end line
line ell b premise isSeries( meta fy , Q ) end line
line ell c premise for all meta m indeed [ meta fx ; meta m ] = [ meta fy ; meta m ] end line
line ell d premise object s1 in0 meta fx end line
line ell e because lemma inSeries indeed exist0 meta m indeed 
  object s1 = (o meta m , [ meta fx ; meta m ] ) end line

line ell f because lemma to=f subset helper indeed
  isSeries( meta fy , Q ) imply
  for all meta m indeed [ meta fx ; meta m ] = [ meta fy ; meta m ] imply
  object s1 = (o meta m , [ meta fx ; meta m ] ) imply
  object s1 in0 meta fy end line
line ell g because prop lemma mp2 modus ponens ell f modus ponens ell b modus ponens ell c indeed
  object s1 = (o meta m , [ meta fx ; meta m ] ) imply
  object s1 in0 meta fy end line
because pred lemma exist mp modus ponens ell g modus ponens ell e indeed
  object s1 in0 meta fy end line
line ell big a end block

any term meta m comma meta fx comma meta fy end line

line ell big b because 1rule deduction modus ponens ell big a indeed
  isSeries( meta fx , Q ) imply
  isSeries( meta fy , Q ) imply
  for all meta m indeed [ meta fx ; meta m ] = [ meta fy ; meta m ] imply
  object s1 in0 meta fx imply
  object s1 in0 meta fy end line

line ell a side condition typeSeries( meta fx , Q ) end line
line ell b side condition typeSeries( meta fy , Q ) end line
line ell c premise for all meta m indeed [ meta fx ; meta m ] = [ meta fy ; meta m ] end line
line ell d because axiom seriesType modus probans ell a indeed isSeries( meta fx , Q ) end line
line ell e because axiom seriesType modus probans ell b indeed isSeries( meta fy , Q ) end line

line ell f because prop lemma mp3 modus ponens ell big b modus ponens ell d modus ponens ell e 
  modus ponens ell c indeed object s1 in0 meta fx imply object s1 in0 meta fy end line
line ell g because 1rule gen modus ponens ell f indeed for all object s1 indeed 
  ( object s1 in0 meta fx imply object s1 in0 meta fy ) end line
because 1rule repetition modus ponens ell g indeed isSubset( meta fx , meta fy ) end line ""; XXREP isSubset
end math ]"


"[  math in theory system Q lemma lemma to=f says
for all terms meta m comma meta fx comma meta fy indeed 
typeSeries( meta fx , Q ) endorse
typeSeries( meta fy , Q ) endorse
for all meta m indeed [ meta fx ; meta m ] = [ meta fy ; meta m ] infer
meta fx = meta fy 
end lemma end math ]"

"[ math system Q proof of lemma to=f reads
any term meta m comma meta fx comma meta fy end line
line ell a side condition typeSeries( meta fx , Q ) end line
line ell b side condition typeSeries( meta fy , Q ) end line
line ell c premise for all meta m indeed [ meta fx ; meta m ] = [ meta fy ; meta m ] end line

line ell d because lemma a4 at meta m modus ponens ell c indeed 
  [ meta fx ; meta m ] = [ meta fy ; meta m ] end line
line ell e because lemma eqSymmetry modus ponens ell d indeed
  [ meta fy ; meta m ] = [ meta fx ; meta m ] end line
line ell f because 1rule gen modus ponens ell e indeed 
  for all meta m indeed [ meta fy ; meta m ] = [ meta fx ; meta m ] end line

line ell g because lemma to=f subset modus probans ell a modus probans ell b modus ponens ell c indeed
  isSubset( meta fx , meta fy ) end line
line ell h because lemma to=f subset modus probans ell b modus probans ell a modus ponens ell f indeed
  isSubset( meta fy , meta fx ) end line
because lemma set equality suff condition modus ponens ell g modus ponens ell h indeed 
  meta fx = meta fy qed
end math ]"

"[  math in theory system Q lemma tester1 says
for all terms meta m comma meta fx comma meta fy indeed ""; XX lidt grimt med den ekstra variabel
for all meta m indeed [ meta fx ; meta m ] = [ meta fy ; meta m ] infer
meta fx = meta fy
end lemma end math ]"

"[ math system Q proof of tester1 reads
any term meta m comma meta fx comma meta fy end line

line ell f premise
  for all meta m indeed [ meta fx  ; meta m ] = [ meta fy ; meta m ] end line

because lemma to=f modus ponens ell f indeed meta fx  = meta fy qed
end math ]"


"[  math in theory system Q lemma lemma positiveNonzero says
for all terms meta x indeed
0 < meta x infer meta x != 0 
end lemma end math ]"

"[ math system Q proof of lemma positiveNonzero reads
any term meta x end line
line ell a premise 0 < meta x end line
line ell b because 1rule repetition modus ponens ell a indeed 
  0 <= meta x and0 0 != meta x end line ""; XXREP <
line ell c because prop lemma second conjunct modus ponens ell b indeed 0 != meta x end line

because lemma neqSymmetry modus ponens ell c indeed meta x != 0 qed
end math ]"

"[  math in theory system Q lemma lemma reciprocalToLeft(Less) says
for all terms meta x comma meta y comma meta z indeed
0 < meta z infer meta x < meta y * 1/ meta z infer
meta x * meta z < meta y
end lemma end math ]"

"[ math system Q proof of lemma reciprocalToLeft(Less) reads
any term meta x comma meta y comma meta z end line
line ell a premise 0 < meta z end line
line ell b premise meta x < meta y * 1/ meta z end line
line ell c because lemma lessMultiplication modus ponens ell a modus ponens ell b indeed
  meta x * meta z < meta y * 1/ meta z * meta z end line


line ell d because lemma three2threeFactors indeed 
  meta y * 1/ meta z * meta z = meta y * meta z * 1/ meta z end line

line ell big a because lemma positiveNonzero modus ponens ell a indeed meta z != 0 end line

line ell e because lemma x=x*y*(1/y) modus ponens ell big a indeed 
  meta y = meta y * meta z * 1/ meta z end line
line ell f because lemma eqSymmetry  modus ponens ell e indeed meta y * meta z * 1/ meta z = meta y end line

line ell g because lemma eqTransitivity modus ponens ell d modus ponens ell f indeed
  meta y * 1/ meta z * meta z = meta y end line
because lemma subLessRight modus ponens ell g modus ponens ell c indeed 
  meta x * meta z < meta y qed
end math ]"




XX 0<1/2 er et specialtilfaelde
"[  math in theory system Q lemma lemma positiveInverted says
for all terms meta x indeed 0 < meta x infer 0 < 1/ meta x
end lemma end math ]"

"[ math system Q proof of lemma positiveInverted reads
any term meta x end line
line ell a premise 0 < meta x end line
line ell b because prop lemma first conjunct modus ponens ell a indeed 0 <= meta x end line
line ell c because prop lemma second conjunct modus ponens ell a indeed 0 != meta x end line
line ell d because lemma neqSymmetry modus ponens ell c indeed meta x != 0 end line
line ell e because lemma 0<1 indeed 0 < 1 end line
line ell f because lemma x*0=0 indeed meta x * 0 = 0 end line
line ell g because lemma x*y=zBackwards  modus ponens ell f indeed 0 = 0 * meta x end line
line ell h because lemma subLessLeft modus ponens ell g modus ponens ell e indeed 
  0 * meta x < 1 end line
line ell i because lemma reciprocal modus ponens ell d indeed meta x * 1/ meta x = 1 end line
line ell j because lemma x*y=zBackwards modus ponens ell i indeed 1 = 1/ meta x * meta x end line
line ell k because lemma subLessRight modus ponens ell j modus ponens ell h indeed
  0 * meta x < 1/ meta x * meta x end line
because lemma lessDivision modus ponens ell b modus ponens ell k indeed 0 < 1/ meta x qed
end math ]"

--------------------

"[  math in theory system Q lemma lemma toNumericalLess says
for all terms meta x comma meta y indeed
- meta y < meta x infer meta x < meta y infer | meta x | < meta y
end lemma end math ]"

"[ math system Q proof of lemma toNumericalLess reads
block any term meta x comma meta y end line
line ell a premise meta x < meta y end line
line ell b premise 0 <= meta x end line
line ell c because lemma nonnegativeNumerical modus ponens ell b indeed | meta x | = meta x end line
line ell d because lemma eqSymmetry modus ponens ell c indeed meta x = | meta x | end line
because lemma subLessLeft modus ponens ell d modus ponens ell a indeed | meta x | < meta y end line
line ell big a end block

block any term meta x comma meta y end line
line ell a premise - meta y < meta x end line
line ell b premise meta x <= 0 end line
line ell e because lemma lessNegated modus ponens ell a indeed - meta x < - - meta y end line

line ell c because lemma nonpositiveNumerical modus ponens ell b indeed | meta x | = - meta x end line
line ell d because lemma eqSymmetry modus ponens ell c indeed - meta x = | meta x | end line
line ell f because lemma subLessLeft modus ponens ell d modus ponens ell e indeed 
  | meta x | < - - meta y end line

line ell g because lemma doubleMinus indeed - - meta y = meta y end line
because lemma subLessRight modus ponens ell g modus ponens ell f indeed | meta x | < meta y end line
line ell big b end block

any term meta x comma meta y end line

line ell a because 1rule deduction modus ponens ell big a indeed
  meta x < meta y imply 0 <= meta x imply | meta x | < meta y end line
line ell b because 1rule deduction modus ponens ell big b indeed
  - meta y < meta x imply meta x <= 0 imply | meta x | < meta y end line
  

line ell c premise - meta y < meta x end line
line ell d premise meta x < meta y end line

line ell e because 1rule mp modus ponens ell a modus ponens ell d indeed 
  0 <= meta x imply | meta x | < meta y end line
line ell f because 1rule mp modus ponens ell b modus ponens ell c indeed 
  meta x <= 0 imply | meta x | < meta y end line

because lemma from leqGeq modus ponens ell e modus ponens ell f indeed 
  | meta x | < meta y qed
end math ]"

"[  math in theory system Q lemma lemma x<=|x| says
for all terms meta x indeed
meta x <= | meta x |
end lemma end math ]"

"[ math system Q proof of lemma x<=|x| reads
block any term meta x end line
line ell a premise 0 <= meta x end line
line ell b because lemma nonnegativeNumerical indeed | meta x | = meta x end line
line ell c because lemma eqSymmetry modus ponens ell b indeed meta x = | meta x | end line
because lemma eqLeq modus ponens ell c indeed meta x <= | meta x | end line
line ell big a end block

block any term meta x end line
line ell a premise meta x <= 0 end line
line ell b because lemma 0<=|x| indeed 0 <= | meta x | end line
because lemma leqTransitivity modus ponens ell a modus ponens ell b indeed meta x <= | meta x | end line
line ell big b end block

any term meta x end line
line ell a because 1rule deduction modus ponens ell big a indeed 
  0 <= meta x imply meta x <= | meta x | end line
line ell b because 1rule deduction modus ponens ell big b indeed 
  meta x <= 0 imply meta x <= | meta x | end line

because lemma from leqGeq modus ponens ell a modus ponens ell b indeed meta x <= | meta x | qed
end math ]"

"[  math in theory system Q lemma lemma positiveToLeft(Less) says
for all terms meta x comma meta y comma meta z indeed
meta x < meta y + meta z infer meta x - meta z < meta y
end lemma end math ]"

"[ math system Q proof of lemma positiveToLeft(Less) reads
any term meta x comma meta y comma meta z end line
line ell a premise meta x < meta y + meta z end line
line ell b because lemma lessAddition modus ponens ell a indeed 
  meta x - meta z < meta y + meta z - meta z end line
line ell c because lemma x=x+y-y indeed meta y = meta y + meta z - meta z end line
line ell d because lemma eqSymmetry modus ponens ell c indeed meta y + meta z - meta z = meta y end line

because lemma subLessRight modus ponens ell d modus ponens ell b indeed meta x - meta z < meta y qed
end math ]"

"[  math in theory system Q lemma lemma negativeToRight(Less) says
for all terms meta x comma meta y comma meta z indeed
meta x - meta y < meta z infer meta x < meta z + meta y 
end lemma end math ]"

"[ math system Q proof of lemma negativeToRight(Less) reads
any term meta x comma meta y comma meta z end line
line ell a premise meta x - meta y < meta z end line
line ell b because lemma lessAddition modus ponens ell a indeed 
  meta x - meta y + meta y < meta z + meta y end line

line ell c because lemma three2threeTerms indeed 
  meta x - meta y + meta y = meta x + meta y - meta y end line
line ell d because lemma x=x+y-y indeed meta x = meta x + meta y - meta y end line
line ell e because lemma eqSymmetry modus ponens ell d indeed meta x + meta y - meta y = meta x end line
line ell f because lemma eqTransitivity modus ponens ell c modus ponens ell e indeed
  meta x - meta y + meta y = meta x end line

because lemma subLessLeft modus ponens ell f modus ponens ell b indeed 
  meta x < meta z + meta y qed
end math ]"


"[  math in theory system Q lemma lemma numericalDifferenceLess helper says
for all terms meta x comma meta y comma meta z indeed
0 <= meta x - meta y infer  meta x - meta y < meta z infer
meta y - meta z < meta x and0 meta x < meta y + meta z 
end lemma end math ]"

"[ math system Q proof of lemma numericalDifferenceLess helper reads
any term meta x comma meta y comma meta z end line
line ell a premise 0 <= meta x - meta y end line
line ell b premise meta x - meta y < meta z end line

line ell c because lemma leqLessTransitivity modus ponens ell a modus ponens ell b indeed 
  0 < meta z end line
line ell d because lemma positiveNegated modus ponens ell c indeed - meta z < 0 end line
line ell e because lemma lessAdditionLeft modus ponens ell d indeed meta y - meta z < meta y + 0 end line
line ell f because axiom plus0 indeed meta y + 0 = meta y end line
line ell g because lemma subLessRight modus ponens ell f modus ponens ell e indeed 
  meta y - meta z < meta y end line
line ell h because lemma negativeToLeft(Leq)(1 term) modus ponens ell a indeed meta y <= meta x end line
line ell i because lemma lessLeqTransitivity modus ponens ell g modus ponens ell h indeed
  meta y - meta z < meta x end line

line ell j because lemma negativeToRight(Less) modus ponens ell b indeed meta x < meta z + meta y end line
line ell k because axiom plusCommutativity indeed meta z + meta y = meta y + meta z end line
line ell l because lemma subLessRight modus ponens ell k modus ponens ell j indeed
  meta x < meta y + meta z end line

because prop lemma join conjuncts modus ponens ell i modus ponens ell l indeed 
 meta y - meta z < meta x and0 meta x < meta y + meta z qed
end math ]"



"[  math in theory system Q lemma lemma numericalDifferenceLess says
for all terms meta x comma meta y comma meta z indeed
| meta x - meta y | < meta z infer
meta y - meta z < meta x and0 meta x < meta y + meta z 
end lemma end math ]"

"[ math system Q proof of lemma numericalDifferenceLess reads
block any term meta x comma meta y comma meta z end line
line ell a premise | meta x - meta y | < meta z end line
line ell b premise 0 <= meta x - meta y end line
line ell c because lemma nonnegativeNumerical modus ponens ell b indeed
  | meta x - meta y | = meta x - meta y end line
line ell d because lemma subLessLeft modus ponens ell c modus ponens ell a indeed
  meta x - meta y < meta z end line
because lemma numericalDifferenceLess helper modus ponens ell b modus ponens ell d indeed
  meta y - meta z < meta x and0 meta x < meta y + meta z end line
line ell big a end block

block any term meta x comma meta y comma meta z end line
line ell a premise | meta x - meta y | < meta z end line
line ell b premise not0 0 <= meta x - meta y end line
line ell c because lemma toLess modus ponens ell b indeed meta x - meta y < 0 end line

line ell d because lemma negativeNumerical modus ponens ell c indeed
  | meta x - meta y | = - parenthesis meta x - meta y end parenthesis end line
line ell e because lemma minusNegated indeed 
  - parenthesis meta x - meta y end parenthesis = meta y - meta x end line
line ell f because lemma eqTransitivity modus ponens ell d modus ponens ell e indeed
  | meta x - meta y | = meta y - meta x end line
line ell g because lemma subLessLeft modus ponens ell f modus ponens ell a indeed
  meta y - meta x < meta z end line

line ell h because lemma negativeNegated modus ponens ell c indeed 
  0 < - parenthesis meta x - meta y end parenthesis end line
line ell i because lemma subLessRight modus ponens ell e modus ponens ell h indeed
  0 < meta y - meta x end line
line ell j because lemma lessLeq modus ponens ell i indeed 0 <= meta y - meta x end line

line ell k because lemma numericalDifferenceLess helper modus ponens ell j modus ponens ell g indeed
  meta x - meta z < meta y and0 meta y < meta x + meta z end line  
line ell l because prop lemma first conjunct modus ponens ell k indeed
  meta x - meta z < meta y end line
line ell m because lemma negativeToRight(Less) modus ponens ell l indeed
  meta x < meta y + meta z end line

line ell n because prop lemma second conjunct modus ponens ell k indeed
  meta y < meta x + meta z end line
line ell o because lemma positiveToLeft(Less) modus ponens ell n indeed meta y - meta z < meta x end line
because prop lemma join conjuncts modus ponens ell o modus ponens ell m indeed
  meta y - meta z < meta x and0 meta x < meta y + meta z end line
line ell big b end block

any term meta x comma meta y comma meta z end line
line ell a because 1rule deduction modus ponens ell big a indeed | meta x - meta y | < meta z imply
  0 <= meta x - meta y imply meta y - meta z < meta x and0 meta x < meta y + meta z end line
line ell b because 1rule deduction modus ponens ell big b indeed | meta x - meta y | < meta z imply
  not0 0 <= meta x - meta y imply meta y - meta z < meta x and0 meta x < meta y + meta z end line

line ell c premise | meta x - meta y | < meta z end line

line ell d because 1rule mp modus ponens ell a modus ponens ell c indeed
  0 <= meta x - meta y imply meta y - meta z < meta x and0 meta x < meta y + meta z end line
line ell e because 1rule mp modus ponens ell b modus ponens ell c indeed
  not0 0 <= meta x - meta y imply meta y - meta z < meta x and0 meta x < meta y + meta z end line

because prop lemma from negations modus ponens ell d modus ponens ell e indeed 
  meta y - meta z < meta x and0 meta x < meta y + meta z qed
end math ]"


"[  math in theory system Q lemma lemma f-Bounded helper says
for all terms meta x comma meta y comma meta z indeed
| meta x - meta y | < meta z infer
| meta y | < max( | meta x + meta z | , | meta x - meta z | )
end lemma end math ]"

"[ math system Q proof of lemma f-Bounded helper reads
any term meta x comma meta y comma meta z end line
line ell a premise | meta x - meta y | < meta z end line

line ell b because lemma numericalDifferenceLess modus ponens ell a indeed
  meta y - meta z < meta x and0 meta x < meta y + meta z end line

line ell c because prop lemma first conjunct modus ponens ell b indeed 
 meta y - meta z < meta x end line
line ell d because lemma negativeToRight(Less) modus ponens ell c indeed
  meta y < meta x + meta z end line
line ell e because lemma x<=|x| indeed meta x + meta z <= | meta x + meta z | end line
line ell f because lemma lessLeqTransitivity modus ponens ell d modus ponens ell e indeed
  meta y < | meta x + meta z | end line 
line ell g because lemma leqMax1 indeed 
  | meta x + meta z | <= max( | meta x + meta z | , | meta x - meta z | ) end line 
line ell h because lemma lessLeqTransitivity modus ponens ell f modus ponens ell g indeed
  meta y < max( | meta x + meta z | , | meta x - meta z | ) end line

line ell i because prop lemma second conjunct modus ponens ell b indeed
  meta x < meta y + meta z end line
line ell j because lemma positiveToLeft(Less) modus ponens ell i indeed
  meta x - meta z < meta y end line

line ell k because lemma x<=|x| indeed meta z - meta x <= | meta z - meta x | end line
line ell l because lemma numericalDifference indeed | meta z - meta x | = | meta x - meta z | end line
line ell m because lemma subLeqRight modus ponens ell l modus ponens ell k indeed
  meta z - meta x <= | meta x - meta z | end line
line ell n because lemma leqNegated modus ponens ell m indeed 
  - | meta x - meta z | <= - parenthesis meta z - meta x end parenthesis end line
line ell o because lemma minusNegated indeed 
  - parenthesis meta z - meta x end parenthesis = meta x - meta z end line
line ell p because lemma subLeqRight modus ponens ell o modus ponens ell n indeed
  - | meta x - meta z | <= meta x - meta z end line
line ell q because lemma leqLessTransitivity modus ponens ell p modus ponens ell j indeed
   - | meta x - meta z | < meta y end line
line ell r because lemma leqMax2 indeed 
  | meta x - meta z | <=  max( | meta x + meta z | , | meta x - meta z | ) end line
line ell s because lemma leqNegated modus ponens ell r indeed
  - max( | meta x + meta z | , | meta x - meta z | ) <= - | meta x - meta z | end line
line ell t because lemma leqLessTransitivity modus ponens ell s modus ponens ell q indeed
  - max( | meta x + meta z | , | meta x - meta z | ) < meta y  end line


because lemma toNumericalLess modus ponens ell t modus ponens ell h indeed 
  | meta y | <  max( | meta x + meta z | , | meta x - meta z | ) qed
end math ]"

"[  math in theory system Q lemma lemma f-Bounded says ""; XX lidt grimt med ekstra variable
for all terms meta v1 comma meta v2 comma meta n comma meta ep comma meta fx indeed
  exist0 meta v1 indeed for all meta v2 indeed | [ meta fx ; meta v2 ] | < meta v1  end lemma end math ]"

"[ math system Q proof of lemma f-Bounded reads
block any term meta v1 comma meta v2 comma meta n comma meta fx end line

line ell a premise for all meta v1 comma meta v2 indeed parenthesis ""; den ubegraensede blok
  0 < 1 imply meta n <= meta v1 imply meta n <= meta v2 imply 
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < 1 end parenthesis end line

line ell big b premise meta n <= meta v2 end line

line ell b because lemma a4 at meta n modus ponens ell a indeed
  for all meta v2 indeed parenthesis
0 < 1 imply meta n <= meta n imply meta n <= meta v2 imply 
| [ meta fx ; meta n ] - [ meta fx ; meta v2 ] | < 1 end parenthesis end line

line ell c because lemma a4 at meta v2 modus ponens ell b indeed
  0 < 1 imply meta n <= meta n imply meta n <= meta v2 imply 
| [ meta fx ; meta n ] - [ meta fx ; meta v2 ] | < 1 end line

line ell d because lemma 0<1 indeed 0 < 1 end line

line ell e because axiom leqReflexivity indeed meta n <= meta n end line

line ell f because prop lemma mp3 modus ponens ell c modus ponens ell d modus ponens ell e
  modus ponens ell big b indeed | [ meta fx ; meta n ] - [ meta fx ; meta v2 ] | < 1 end line

because lemma f-Bounded helper modus ponens ell f indeed
  | [ meta fx ; meta v2 ] | < max( | [ meta fx ; meta n ] + 1 | , | [ meta fx ; meta n ] -  1 | ) end line

line ell big a end block

block any term meta v1 comma meta v2 comma meta n comma meta fx end line
line ell b premise for all meta v2 indeed 
  parenthesis meta v2 <= meta n imply | [ meta fx ; meta v2 ] | < meta v1 end parenthesis end line
line ell a premise not0 meta n <= meta v2 end line

line ell c because lemma a4 at meta v2 modus ponens ell b indeed
  meta v2 <= meta n imply | [ meta fx ; meta v2 ] | < meta v1 end line

line ell d because lemma toLess modus ponens ell a indeed meta v2 < meta n end line
line ell e because lemma lessLeq modus ponens ell d indeed meta v2 <= meta n end line

because 1rule mp modus ponens ell c modus ponens ell e indeed
  | [ meta fx ; meta v2 ] | < meta v1 end line
line ell big b end block

any term meta v1 comma meta v2 comma meta n comma meta ep comma meta fx end line

line ell a because 1rule deduction modus ponens ell big a indeed
  for all meta v1 comma meta v2 indeed parenthesis
  0 < 1 imply meta n <= meta v1 imply meta n <= meta v2 imply 
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < 1 end parenthesis  imply
  meta n <= meta v2 imply
  | [ meta fx ; meta v2 ] | < max( | [ meta fx ; meta n ] + 1 | , | [ meta fx ; meta n ] -  1 | ) end line

line ell b because 1rule deduction modus ponens ell big b indeed
   for all meta v2 indeed parenthesis meta v2 <= meta n imply | [ meta fx ; meta v2 ] | < meta v1 
   end parenthesis imply not0 meta n <= meta v2 imply
    | [ meta fx ; meta v2 ] | < meta v1 end line


line ell c because axiom cauchy indeed 
  for all meta ep indeed exist0 meta n indeed for all meta v1 comma meta v2 indeed parenthesis
  0 < meta ep imply meta n <= meta v1 imply meta n <= meta v2 imply 
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < meta ep end parenthesis end line

line ell d because lemma a4 at 1 modus ponens ell c indeed
  exist0 meta n indeed for all meta v1 comma meta v2 indeed parenthesis
  0 < 1 imply meta n <= meta v1 imply meta n <= meta v2 imply 
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < 1 end parenthesis end line


line ell e because lemma fpart-Bounded indeed
  exist0 meta v1 indeed for all meta v2 indeed 
  parenthesis meta v2 <= meta n imply | [ meta fx ; meta v2 ] | < meta v1 end parenthesis end line

line ell f because pred lemma exist mp modus ponens ell a modus ponens ell d indeed
  meta n <= meta v2 imply  
  | [ meta fx ; meta v2 ] | < max( | [ meta fx ; meta n ] + 1 | , | [ meta fx ; meta n ] - 1 | ) end line
line ell g because pred lemma exist mp modus ponens ell b modus ponens ell e indeed
  not0 meta n <= meta v2 imply | [ meta fx ; meta v2 ] | < meta v1 end line

line ell h because lemma lessThanMax modus ponens ell f modus ponens ell g indeed
  | [ meta fx ; meta v2 ] | < 
  max( max( | [ meta fx ; meta n ] + 1 | , | [ meta fx ; meta n ] - 1 | ) , meta v1 ) end line

line ell i because 1rule gen modus ponens ell h indeed for all meta v2 indeed 
  | [ meta fx ; meta v2 ] | <
  max( max( | [ meta fx ; meta n ] + 1 | , | [ meta fx ; meta n ] - 1 | ) , meta v1 ) end line

because pred lemma intro exist 
  at max( max( | [ meta fx ; meta n ] + 1 | , | [ meta fx ; meta n ] - 1 | ) , meta v1 ) 
  modus ponens ell i indeed 
exist0 meta v1 indeed for all meta v2 indeed | [ meta fx ; meta v2 ] | < meta v1 qed
end math ]"

"[  math in theory system Q lemma lemma fpart-Bounded base says
for all terms meta v1 comma meta v2n comma meta fx indeed
  exist0 meta v1 indeed for all meta v2n indeed 
  parenthesis meta v2n <= 0 imply | [ meta fx ; meta v2n ] | < meta v1 end parenthesis 
end lemma end math ]"

"[ math system Q proof of lemma fpart-Bounded base reads
block any term meta v1 comma meta v2n comma meta fx end line
line ell b premise meta v2n <= 0 end line
line ell c because lemma leqLessEq modus ponens ell b indeed meta v2n < 0 or0 meta v2n = 0 end line
line ell d because axiom nonnegative(N) indeed 0 <= meta v2n end line
line ell big d because lemma toNotLess modus ponens ell d indeed not0 meta v2n < 0 end line
line ell e because prop lemma negate first disjunct modus ponens ell c modus ponens ell big d indeed 
  meta v2n = 0 end line
line ell f because lemma sameSeries modus ponens ell e indeed 
  [ meta fx ; meta v2n ] = [ meta fx ; 0 ] end line
line ell g because lemma sameNumerical modus ponens ell f indeed
  | [ meta fx ; meta v2n ] | = | [ meta fx ; 0 ] | end line
line ell h because lemma eqAddition modus ponens ell g indeed
  | [ meta fx ; meta v2n ] | + 1 = | [ meta fx ; 0 ] | + 1 end line

line ell i because axiom leqReflexivity indeed 
  | [ meta fx ; meta v2n ] | <= | [ meta fx ; meta v2n ] | end line
line ell j because lemma leqPlus1 modus ponens ell i indeed
  | [ meta fx ; meta v2n ] | < | [ meta fx ; meta v2n ] | + 1 end line

because lemma subLessRight modus ponens ell h modus ponens ell j indeed 
  | [ meta fx ; meta v2n ] | < | [ meta fx ; 0 ] | + 1 end line
line ell big a end block


any term meta v1 comma meta v2n comma meta fx end line

line ell a because 1rule deduction modus ponens ell big a indeed
  meta v2n <= 0 imply | [ meta fx ; meta v2n ] | < | [ meta fx ; 0 ] | + 1 end line

line ell b because 1rule gen modus ponens ell a indeed
  for all meta v2n indeed parenthesis meta v2n <= 0 imply 
  | [ meta fx ; meta v2n ] | < | [ meta fx ; 0 ] | + 1 end parenthesis end line

because pred lemma intro exist at | [ meta fx ; 0 ] | + 1 modus ponens ell b indeed 
  exist0 meta v1 indeed for all meta v2n indeed parenthesis meta v2n <= 0 imply 
  | [ meta fx ; meta v2n ] | < meta v1 end parenthesis qed
end math ]"




"[  math in theory system Q lemma lemma fpart-Bounded indu helper says
for all terms meta v1 comma meta v2n comma meta n comma meta fx indeed
meta v2n <= meta n imply | [ meta fx ; meta v2n ] | < meta v1 infer
meta v2n <= meta n + 1 infer | [ meta fx ; meta v2n ] | < max( meta v1 , | [ meta fx ; meta n + 1 ] | + 1 )
end lemma end math ]"

"[ math system Q proof of lemma fpart-Bounded indu helper reads
any term meta v1 comma meta v2n comma meta n comma meta fx end line
line ell c premise meta v2n <= meta n imply | [ meta fx ; meta v2n ] | < meta v1 end line
line ell d premise meta v2n <= meta n + 1 end line
line ell e because lemma leqLessEq modus ponens ell d indeed 
  meta v2n < meta n + 1 or0 meta v2n = meta n + 1 end line

block any term meta v1 comma meta v2n comma meta n comma meta fx end line
line ell c premise meta v2n <= meta n imply | [ meta fx ; meta v2n ] | < meta v1 end line
line ell d premise meta v2n < meta n + 1 end line
line ell e because 1rule lessMinus1(N) modus ponens ell d indeed
  meta v2n <= meta n end line

line ell big e because 1rule mp modus ponens ell c modus ponens ell e indeed 
  | [ meta fx ; meta v2n ] | < meta v1 end line
line ell big f because lemma leqMax1 indeed 
  meta v1 <= max( meta v1 , | [ meta fx ; meta n + 1 ] | + 1 ) end line 

because lemma lessLeqTransitivity modus ponens ell big e modus ponens ell big f indeed 
  | [ meta fx ; meta v2n ] | < max( meta v1 , | [ meta fx ; meta n + 1 ] | + 1 ) end line
line ell big a end block

block any term meta v1 comma meta v2n comma meta n comma meta fx end line 
line ell b premise meta v2n = meta n + 1 end line
line ell c because lemma sameSeries modus ponens ell b indeed
  [ meta fx ; meta v2n ]  = [ meta fx ; meta n + 1 ] end line
line ell d because lemma sameNumerical modus ponens ell c indeed
  | [ meta fx ; meta v2n ] | = | [ meta fx ; meta n + 1 ] | end line

line ell e because lemma eqLeq modus ponens ell d indeed
  | [ meta fx ; meta v2n ] | <= | [ meta fx ; meta n + 1 ] | end line
line ell f because lemma leqPlus1 modus ponens ell e indeed
  | [ meta fx ; meta v2n ] | < | [ meta fx ; meta n + 1 ] | + 1 end line
line ell g because lemma leqMax2 indeed 
  | [ meta fx ; meta n + 1 ] | + 1 <= max( meta v1 , | [ meta fx ; meta n + 1 ] | + 1 ) end line
because lemma lessLeqTransitivity modus ponens ell f modus ponens ell g indeed 
  | [ meta fx ; meta v2n ] | < max( meta v1 , | [ meta fx ; meta n + 1 ] | + 1 ) end line
line ell big b end block


line ell f because 1rule deduction modus ponens ell big a indeed  
  parenthesis meta v2n <= meta n imply | [ meta fx ; meta v2n ] | < meta v1 end parenthesis imply
  meta v2n < meta n + 1 imply | [ meta fx ; meta v2n ] | < max( meta v1 , | [ meta fx ; meta n + 1 ] | + 1 ) 
  end line

line ell g because 1rule mp modus ponens ell f modus ponens ell c indeed meta v2n < meta n + 1 imply 
  | [ meta fx ; meta v2n ] | < max( meta v1 , | [ meta fx ; meta n + 1 ] | + 1 ) end line

line ell i because 1rule deduction modus ponens ell big b indeed  
  meta v2n = meta n + 1 imply | [ meta fx ; meta v2n ] | < max( meta v1 , | [ meta fx ; meta n + 1 ] | + 1 ) 
  end line

because prop lemma from disjuncts modus ponens ell e modus ponens ell g modus ponens ell i indeed
  | [ meta fx ; meta v2n ] | < max( meta v1 , | [ meta fx ; meta n + 1 ] | + 1 ) qed
end math ]"

"[  math in theory system Q lemma lemma fpart-Bounded indu says
for all terms meta v1 comma meta v2n comma meta n comma meta fx indeed

exist0 meta v1 indeed for all meta v2n indeed parenthesis meta v2n <= meta n imply 
| [ meta fx ; meta v2n ] | < meta v1 end parenthesis imply
exist0 meta v1 indeed for all meta v2n indeed parenthesis meta v2n <= meta n + 1 imply 
| [ meta fx ; meta v2n ] | < meta v1 end parenthesis 
end lemma end math ]"


"[ math system Q proof of lemma fpart-Bounded indu reads
block any term meta v1 comma meta v2n comma meta n comma meta fx end line
line ell c premise meta v2n <= meta n imply | [ meta fx ; meta v2n ] | < meta v1 end line
line ell d premise meta v2n <= meta n + 1 end line
because lemma fpart-Bounded indu helper modus ponens ell c modus ponens ell d indeed 
  | [ meta fx ; meta v2n ] | < max( meta v1 , | [ meta fx ; meta n + 1 ] | + 1 ) end line
line ell big a end block

any term meta v1 comma meta v2n comma meta n comma meta fx end line
line ell a because 1rule deduction modus ponens ell big a indeed
  parenthesis meta v2n <= meta n imply | [ meta fx ; meta v2n ] | < meta v1   
  end parenthesis imply parenthesis meta v2n <= meta n + 1 imply
  | [ meta fx ; meta v2n ] | < max( meta v1 , | [ meta fx ; meta n + 1 ] | + 1 ) end parenthesis end line

line ell b because pred lemma addAll modus ponens ell a indeed
  for all meta v2n indeed parenthesis meta v2n <= meta n imply | [ meta fx ; meta v2n ] | < meta v1   
  end parenthesis imply 
  for all meta v2n indeed parenthesis meta v2n <= meta n + 1 imply
  | [ meta fx ; meta v2n ] | < max( meta v1 , | [ meta fx ; meta n + 1 ] | + 1 ) end parenthesis end line

because pred lemma addExist(SimpleAnt) at max( meta v1 , | [ meta fx ; meta n + 1 ] | + 1 ) 
  modus ponens ell b indeed 
  exist0 meta v1 indeed for all meta v2n indeed 
  parenthesis meta v2n <= meta n imply | [ meta fx ; meta v2n ] | < meta v1 end parenthesis imply 
  exist0 meta v1 indeed for all meta v2n indeed parenthesis meta v2n <= meta n + 1 imply
  | [ meta fx ; meta v2n ] | < meta v1 end parenthesis qed

end math ]"




"[  math in theory system Q lemma lemma fpart-Bounded says
for all terms meta v1 comma meta v2 comma meta n comma meta fx indeed
  exist0 meta v1 indeed for all meta v2 indeed 
  parenthesis meta v2 <= meta n imply | [ meta fx ; meta v2 ] | < meta v1 end parenthesis
end lemma end math ]"

"[ math system Q proof of lemma fpart-Bounded reads
any term meta v1 comma meta v2 comma meta n comma meta fx end line
line ell a because lemma fpart-Bounded base indeed exist0 meta v1 indeed for all meta v2 indeed
  parenthesis meta v2 <= 0 imply | [ meta fx ; meta v2 ] | < meta v1 end parenthesis end line

line ell b because lemma fpart-Bounded indu indeed exist0 meta v1 indeed for all meta v2 indeed
  parenthesis meta v2 <= meta n imply | [ meta fx ; meta v2 ] | < meta v1 end parenthesis imply
  exist0 meta v1 indeed for all meta v2 indeed
  parenthesis meta v2 <= meta n + 1 imply | [ meta fx ; meta v2 ] | < meta v1 end parenthesis end line

because lemma induction modus ponens ell a modus ponens ell b indeed 
  exist0 meta v1 indeed for all meta v2 indeed parenthesis 
  meta v2 <= meta n imply | [ meta fx ; meta v2 ] | < meta v1 end parenthesis qed
end math ]"



"[  math in theory system Q lemma lemma negativeToLeft(Leq)(1 term) says 
for all terms meta y comma meta z indeed
0 <= meta y - meta z infer meta z <= meta y  
end lemma end math ]"

"[ math system Q proof of lemma negativeToLeft(Leq)(1 term) reads
any term meta y comma meta z end line
line ell a premise 0 <= meta y - meta z end line
line ell b because lemma negativeToLeft(Leq) modus ponens ell a indeed 0 + meta z <= meta y end line
line ell c because lemma plus0Left indeed 0 + meta z = meta z end line
because lemma subLeqLeft modus ponens ell c indeed meta z <=  meta y qed
end math ]"





"[  math in theory system Q lemma lemma negativeToLeft(Less) says
for all terms meta x comma meta y comma meta z indeed
meta x < meta y - meta z infer meta x + meta z < meta y
end lemma end math ]"

"[ math system Q proof of lemma negativeToLeft(Less) reads
any term meta x comma meta y comma meta z end line
line ell a premise meta x < meta y - meta z end line
line ell b because lemma lessAddition modus ponens ell a indeed 
  meta x + meta z < meta y - meta z + meta z end line

line ell c because lemma three2threeTerms indeed meta y - meta z + meta z = meta y + meta z - meta z end line
line ell d because lemma x=x+y-y indeed meta y = meta y + meta z - meta z end line
line ell e because lemma eqSymmetry modus ponens ell d indeed meta y + meta z - meta z = meta y end line
line ell f because lemma eqTransitivity modus ponens ell c modus ponens ell e indeed
  meta y - meta z + meta z = meta y end line

because lemma subLessRight modus ponens ell f modus ponens ell b indeed meta x + meta z < meta y qed
end math ]"



"[ math in theory system Q lemma lemma positiveTripled says
for all terms meta x indeed
0 < 1/3 * meta x infer 0 < meta x  
end lemma end math ]"

"[ math system Q proof of lemma positiveTripled reads
any term meta x end line
line ell a premise 0 < 1/3 * meta x end line
line ell b because lemma 0<3 indeed 0 < 3 end line
line ell c because lemma positiveFactors modus ponens ell b modus ponens ell a indeed 
  0 < 3 * ( 1/3 * meta x ) end line

line ell x because axiom timesAssociativity indeed 3 * 1/3 * meta x = 3 * ( 1/3 * meta x ) end line
line ell y because lemma eqSymmetry modus ponens ell x indeed 
  3 * ( 1/3 * meta x ) = 3 * 1/3 * meta x end line

line ell d because lemma positiveNonzero modus ponens ell b indeed 3 != 0 end line
line ell e because lemma reciprocal modus ponens ell d indeed 3 * 1/3 = 1 end line
line ell f because lemma eqMultiplication modus ponens ell e indeed 
  3 * 1/3 * meta x = 1 * meta x end line

line ell g because lemma times1Left indeed 1 * meta x = meta x end line
line ell h because lemma eqTransitivity4 modus ponens ell y modus ponens ell f modus ponens ell g indeed 
  3 * ( 1/3 * meta x ) = meta x end line
because lemma subLessRight modus ponens ell h modus ponens ell c indeed 0 < meta x qed
end math ]"

"[ math in theory system Q lemma lemma positiveDividedBy3 says
for all terms meta x indeed
0 < meta x infer 0 < 1/3 * meta x
end lemma end math ]"

"[ math system Q proof of lemma positiveDividedBy3 reads
any term meta x end line
line ell a premise 0 < meta x end line
line ell b because lemma 0<3 indeed 0 < 3 end line
line ell c because lemma positiveInverted modus ponens ell b indeed 0 < 1/3 end line
because lemma positiveFactors modus ponens ell c modus ponens ell a indeed
  0 < 1/3 * meta x qed
end math ]"

"[ math in theory system Q lemma lemma |x-x|=0 says
for all terms meta x indeed | meta x - meta x | = 0 
end lemma end math ]"

"[ math system Q proof of lemma |x-x|=0 reads
any term meta x end line
line ell a because lemma eqReflexivity indeed meta x = meta x end line
line ell b because lemma positiveToLeft(Eq)(1 term) modus ponens ell a indeed meta x - meta x = 0 end line
line ell c because lemma sameNumerical modus ponens ell b indeed | meta x - meta x | = | 0 | end line
line ell d because lemma |0|=0 indeed | 0 | = 0 end line
because lemma eqTransitivity modus ponens ell c modus ponens ell d indeed | meta x - meta x | = 0 qed
end math ]"

"[ math in theory system Q lemma lemma 1<2 says
1 < 2 end lemma end math ]"

"[ math system Q proof of lemma 1<2 reads
line ell a because lemma 0<1 indeed 0 < 1 end line
line ell b because lemma lessAddition modus ponens ell a indeed 0 + 1 < 1 + 1 end line
line ell c because lemma plus0Left indeed 0 + 1 = 1 end line
line ell d because lemma subLessLeft modus ponens ell c modus ponens ell b indeed 1 < 1 + 1 end line
because 1rule repetition modus ponens ell d indeed 1 < 2 qed ""; XXREP 2
end math ]"

"[ math in theory system Q lemma lemma 1/3<2/3 says 1/3 < 2/3
end lemma end math ]"

"[ math system Q proof of lemma 1/3<2/3 reads
line ell a because lemma 1<2 indeed 1 < 2 end line
line ell b because lemma 0<1/3 indeed 0 < 1/3 end line
line ell c because lemma lessMultiplication modus ponens ell b modus ponens ell a indeed 
  1 * 1/3 < 2 * 1/3 end line
line ell d because lemma times1Left indeed 1 * 1/3 = 1/3 end line
line ell e because lemma subLessLeft modus ponens ell d modus ponens ell c indeed 1/3 < 2 * 1/3 end line
because 1rule repetition modus ponens ell e indeed 1/3 < 2/3 qed ""; XXREP 2/3
end math ]"



"[ math in theory system Q lemma lemma (1/3)x+(1/3)x=(2/3)x says
for all terms meta x indeed
1/3 * meta x + 1/3 * meta x = 2/3 * meta x
end lemma end math ]"

"[ math system Q proof of lemma (1/3)x+(1/3)x=(2/3)x reads
any term meta x end line
line ell a because lemma x+x=2*x indeed 1/3 * meta x + 1/3 * meta x = 2 * ( 1/3 * meta x ) end line
line ell b because axiom timesAssociativity indeed 2 * 1/3 * meta x = 2 * ( 1/3 * meta x ) end line
line ell c because lemma eqSymmetry modus ponens ell b indeed 
  2 * ( 1/3 * meta x ) = 2 * 1/3 * meta x  end line
line ell d because lemma eqTransitivity modus ponens ell a modus ponens ell c indeed
  1/3 * meta x + 1/3 * meta x = 2 * 1/3 * meta x  end line
because 1rule repetition modus ponens ell d indeed 
  1/3 * meta x + 1/3 * meta x = 2/3 * meta x qed ""; XXREP 2/3
end math ]"


"[ math in theory system Q lemma lemma -(1/3)x-(1/3)x=-(2/3)x says
for all terms meta x indeed
- ( 1/3 * meta x ) - 1/3 * meta x = - ( 2/3 * meta x )
end lemma end math ]"

"[ math system Q proof of lemma -(1/3)x-(1/3)x=-(2/3)x  reads
any term meta x end line
line ell a because lemma (1/3)x+(1/3)x=(2/3)x indeed 1/3 * meta x + 1/3 * meta x = 2/3 * meta x end line
line ell b because lemma eqNegated modus ponens ell a indeed 
  - ( 1/3 * meta x + 1/3 * meta x ) = - ( 2/3 * meta x ) end line
line ell c because lemma -x-y=-(x+y) indeed 
  - ( 1/3 * meta x ) - 1/3 * meta x = - ( 1/3 * meta x + 1/3 * meta x ) end line
because lemma eqTransitivity modus ponens ell c modus ponens ell b indeed 
  - ( 1/3 * meta x ) - 1/3 * meta x = - ( 2/3 * meta x ) qed
end math ]"

"[ math in theory system Q lemma lemma (2/3)x+(1/3)x=x says
for all terms meta x indeed
2/3 * meta x + 1/3 * meta x = meta x 
end lemma end math ]"

"[ math system Q proof of lemma (2/3)x+(1/3)x=x reads
any term meta x end line
line ell x because lemma x+x=2*x indeed 1/3 * meta x + 1/3 * meta x = 2 * ( 1/3 * meta x ) end line
line ell y because axiom timesAssociativity indeed 2 * 1/3 * meta x = 2 * ( 1/3 * meta x ) end line
line ell z because lemma eqSymmetry modus ponens ell y indeed 
  2 * ( 1/3 * meta x ) = 2 * 1/3 * meta x end line
line ell a because lemma eqTransitivity modus ponens ell x modus ponens ell z indeed 
  1/3 * meta x + 1/3 * meta x = 2/3 * meta x end line
line ell b because lemma eqAddition modus ponens ell a indeed
  1/3 * meta x + 1/3 * meta x + 1/3 * meta x = 2/3 * meta x + 1/3 * meta x end line
line ell c because lemma (1/3)x+(1/3)x+(1/3)x=x indeed 
  1/3 * meta x + 1/3 * meta x + 1/3 * meta x = meta x end line
because lemma equality modus ponens ell b modus ponens ell c indeed 2/3 * meta x + 1/3 * meta x = meta x qed
end math ]"


"[ math in theory system Q lemma lemma -x+(1/3)x=-(2/3)x says
for all terms meta x indeed
- meta x + 1/3 * meta x = - ( 2/3 * meta x )
end lemma end math ]"

"[ math system Q proof of lemma -x+(1/3)x=-(2/3)x reads
any term meta x end line
line ell a because lemma (2/3)x+(1/3)x=x indeed 
   2/3 * meta x + 1/3 * meta x = meta x end line
line ell b because lemma positiveToRight(Eq) modus ponens ell a indeed 
  2/3 * meta x = meta x - 1/3 * meta x end line
line ell c because lemma eqNegated modus ponens ell b indeed
  - ( 2/3 * meta x ) = - ( meta x - 1/3 * meta x ) end line
line ell d because lemma minusNegated indeed - ( meta x - 1/3 * meta x ) = 1/3 * meta x - meta x end line
line ell e because axiom plusCommutativity indeed 
  1/3 * meta x - meta x = - meta x + 1/3 * meta x end line

line ell f because lemma eqTransitivity4 modus ponens ell c modus ponens ell d modus ponens ell e indeed
  - ( 2/3 * meta x ) = - meta x + 1/3 * meta x end line
because lemma eqSymmetry modus ponens ell f indeed - meta x + 1/3 * meta x = - ( 2/3 * meta x ) qed
end math ]"

line ell y because lemma (2/3)x+(1/3)x=x indeed 
  2/3 * meta x + 1/3 * meta x = meta x end line
line ell a because lemma eqTransitivity modus ponens ell x modus ponens ell y indeed
  1/3 * meta x + 2/3 * meta x = meta x end line



"[ math in theory system Q lemma lemma preserveLessGreater says
for all terms meta x1 comma meta x2 comma meta y1 comma meta y2 comma meta z indeed
meta x1 <= meta y1 - meta z infer | meta x1 - meta x2 | < 1/3 * meta z  infer
| meta y1 - meta y2 | < 1/3 * meta z  infer
meta x2 <= meta y2 - 1/3 * meta z 
end lemma end math ]"

"[ math system Q proof of lemma preserveLessGreater reads
any term meta x1 comma meta x2 comma meta y1 comma meta y2 comma meta z end line
line ell b premise meta x1 <= meta y1 - meta z end line
line ell c premise | meta x1 - meta x2 | < 1/3 * meta z end line
line ell d premise | meta y1 - meta y2 | < 1/3 * meta z end line

line ell e because lemma 0<=|x| indeed 0 <= | meta x1 - meta x2 | end line
line ell f because lemma leqLessTransitivity modus ponens ell e modus ponens ell c indeed
  0 < 1/3 * meta z end line
line ell x because lemma positiveTripled modus ponens ell f indeed 0 < meta z end line


line ell g because lemma numericalDifferenceLess modus ponens ell c indeed
  meta x2 - 1/3 * meta z < meta x1 and0 meta x1 < meta x2 + 1/3 * meta z end line
line ell h because prop lemma first conjunct modus ponens ell g indeed
  meta x2 - 1/3 * meta z < meta x1 end line
line ell i because lemma negativeToRight(Less) modus ponens ell h indeed
  meta x2 < meta x1 + 1/3 * meta z end line

line ell big j because lemma leqAddition modus ponens ell b indeed
  meta x1 + 1/3 * meta z <= meta y1 - meta z + 1/3 * meta z end line
line ell big k because lemma -x+(1/3)x=-(2/3)x indeed
  - meta z + 1/3 * meta z = - ( 2/3 * meta z ) end line
line ell big l because lemma three2twoTerms modus ponens ell big k indeed
  meta y1 - meta z + 1/3 * meta z = meta y1 - 2/3 * meta z end line
line ell m because lemma subLeqRight modus ponens ell big l modus ponens ell big j indeed
  meta x1 + 1/3 * meta z <= meta y1 - 2/3 * meta z end line

line ell big s because lemma numericalDifferenceLess modus ponens ell d indeed
  meta y2 - 1/3 * meta z < meta y1 and0 meta y1 < meta y2 + 1/3 * meta z end line
line ell big t because prop lemma second conjunct modus ponens ell big s indeed 
  meta y1 < meta y2 + 1/3 * meta z end line
line ell big m because lemma positiveToLeft(Less) modus ponens ell big t indeed
  meta y1 - 1/3 * meta z < meta y2 end line
line ell big n because lemma lessAddition modus ponens ell big m indeed 
  meta y1 - 1/3 * meta z - 1/3 * meta z < meta y2 - 1/3 * meta z end line

line ell big o because axiom plusAssociativity indeed 
  meta y1 - 1/3 * meta z - 1/3 * meta z = meta y1 + ( - ( 1/3 * meta z ) - 1/3 * meta z ) end line
line ell big p because lemma -(1/3)x-(1/3)x=-(2/3)x indeed
  - ( 1/3 * meta z ) - 1/3 * meta z = - ( 2/3 * meta z ) end line
line ell big q because lemma eqAdditionLeft modus ponens ell big p indeed
  meta y1 + ( - ( 1/3 * meta z ) - 1/3 * meta z ) = meta y1 - ( 2/3 * meta z ) end line
line ell big r because lemma eqTransitivity modus ponens ell big o modus ponens ell big q indeed
  meta y1 - 1/3 * meta z - 1/3 * meta z = meta y1 - ( 2/3 * meta z ) end line
line ell p because lemma subLessLeft modus ponens ell big r modus ponens ell big n indeed
  meta y1 - 2/3 * meta z < meta y2 - 1/3 * meta z end line


line ell q because lemma lessLeqTransitivity modus ponens ell i modus ponens ell m indeed
  meta x2 < meta y1 - 2/3 * meta z end line
line ell r because lemma lessTransitivity modus ponens ell q modus ponens ell p indeed
  meta x2 < meta y2 - 1/3 * meta z end line
because lemma lessLeq modus ponens ell r indeed meta x2 <= meta y2 - 1/3 * meta z qed
end math ]"








"[ math in theory system Q lemma lemma closetolessIsLess says
for all terms meta x1 comma meta x2 comma meta y comma meta z indeed
meta x1 <= meta y - meta z infer | meta x1 - meta x2 | < 1/3 * meta z  infer
meta x2 <= meta y - 1/3 * meta z 
end lemma end math ]"

"[ math system Q proof of lemma closetolessIsLess reads
any term meta x1 comma meta x2 comma meta y comma meta z end line
line ell a premise meta x1 <= meta y - meta z end line
line ell b premise | meta x1 - meta x2 | < 1/3 * meta z end line

line ell c because lemma 0<=|x| indeed 0 <= | meta x1 - meta x2 | end line
line ell d because lemma leqLessTransitivity modus ponens ell c modus ponens ell b indeed 
  0 < 1/3 * meta z end line

line ell e because lemma |x-x|=0 indeed | meta y - meta y | = 0 end line
line ell f because lemma eqSymmetry modus ponens ell e indeed 0 = | meta y - meta y | end line
line ell g because lemma subLessLeft modus ponens ell f modus ponens ell d indeed
  | meta y - meta y | < 1/3 * meta z end line

because lemma preserveLessGreater modus ponens ell a modus ponens ell b modus ponens ell g indeed
  meta x2 <= meta y - 1/3 * meta z qed
end math ]"


"[ math in theory system Q lemma lemma subLessLeft(F) says
for all terms meta fx comma meta fy comma meta fz indeed
meta fx sameF meta fy infer meta fx <f meta fz infer meta fy <f meta fz
end lemma end math ]"

"[ math system Q proof of lemma subLessLeft(F) reads
any term meta fx comma meta fy comma meta fz end line
line ell a premise meta fx sameF meta fy end line
line ell b premise meta fx <f meta fz end line
line ell c because 1rule repetition modus ponens ell a indeed ""; XXREP sameF
  for all object ep indeed exist0 object n indeed for all object m indeed (
  0 < object ep imply object n <= object m imply 
  | [ meta fx ; object m ] - [ meta fy ; object m ] | < object ep ) end line

line ell d because 1rule deduction modus ponens ell c indeed ""; XXded
  for all object ep indeed exist0 object n1 indeed for all object m indeed (
  0 < object ep imply object n1 <= object m imply 
  | [ meta fx ; object m ] - [ meta fy ; object m ] | < object ep ) end line

line ell e because lemma a4 at 1/3 * object ep modus ponens ell c indeed
  exist0 object n1 indeed for all object m indeed (
  0 < 1/3 * object ep imply object n1 <= object m imply 
  | [ meta fx ; object m ] - [ meta fy ; object m ] | < 1/3 * object ep ) end line

line ell f because 1rule repetition modus ponens ell b indeed ""; XXREP <f
  exist0 object ep indeed exist0 object n indeed for all object m indeed 0 < object ep and0 
  ( object n <= object m imply [ meta fx ; object m ] <= [ meta fz ; object m ] - object ep ) end line

line ell g because 1rule deduction modus ponens ell f indeed ""; XXded
  exist0 object ep indeed exist0 object n2 indeed for all object m indeed 0 < object ep and0 
  ( object n2 <= object m imply [ meta fx ; object m ] <= [ meta fz ; object m ] - object ep ) end line

""; XX kan godt optimeres til 1 blok, men det kraever 
""; 1. 'intelligente' blokke 2. en aendring af <f-definitionen
block any term meta fx comma meta fx comma meta fz end line
line ell a premise for all object m indeed 0 < object ep and0 ""; XX linie g
  ( object n2 <= object m imply [ meta fx ; object m ] <= [ meta fz ; object m ] - object ep ) end line
line ell b because lemma a4 at object m modus ponens ell a indeed 0 < object ep and0 
  ( object n2 <= object m imply [ meta fx ; object m ] <= [ meta fz ; object m ] - object ep ) end line
line ell c because prop lemma first conjunct modus ponens ell b indeed 0 < object ep end line
because lemma positiveDividedBy3 modus ponens ell c indeed 0 < 1/3 * object ep end line
line ell big a end block


block any term meta fx comma meta fy comma meta fz end line
line ell a premise for all object m indeed ( ""; XX linie e
  0 < 1/3 * object ep imply object n1 <= object m imply 
  | [ meta fx ; object m ] - [ meta fy ; object m ] | < 1/3 * object ep ) end line
line ell b premise for all object m indeed 0 < object ep and0  ""; XX linie g
  ( object n2 <= object m imply [ meta fx ; object m ] <= [ meta fz ; object m ] - object ep ) end line
line ell x premise max( object n1 , object n2 ) <= object m end line
line ell big b because lemma leqMax1 indeed object n1 <= max( object n1 , object n2 ) end line
line ell big c because lemma leqTransitivity modus ponens ell big b modus ponens ell x indeed
  object n1 <= object m end line
line ell big d because lemma leqMax2 indeed object n2 <= max( object n1 , object n2 ) end line
line ell big e because lemma leqTransitivity modus ponens ell big d modus ponens ell x indeed
  object n2 <= object m end line

line ell d because lemma a4 at object m modus ponens ell b indeed 0 < object ep and0
  ( object n2 <= object m imply [ meta fx ; object m ] <= [ meta fz ; object m ] - object ep ) end line
line ell e because prop lemma first conjunct modus ponens ell d indeed 0 < object ep end line
line ell f because prop lemma second conjunct modus ponens ell d indeed 
  object n2 <= object m imply [ meta fx ; object m ] <= [ meta fz ; object m ] - object ep end line
line ell g because 1rule mp modus ponens ell f modus ponens ell big e indeed
  [ meta fx ; object m ] <= [ meta fz ; object m ] - object ep end line

line ell h because lemma a4 at object m modus ponens ell a indeed
  0 < 1/3 * object ep imply object n1 <= object m imply 
  | [ meta fx ; object m ] - [ meta fy ; object m ] | < 1/3 * object ep end line
line ell i because lemma positiveDividedBy3 modus ponens ell e indeed 0 < 1/3 * object ep end line
line ell j because prop lemma mp2 modus ponens ell h modus ponens ell i modus ponens ell big c indeed
  | [ meta fx ; object m ] - [ meta fy ; object m ] | < 1/3 * object ep end line
because lemma closetolessIsLess modus ponens ell g modus ponens ell j indeed
   [ meta fy ; object m ] <= [ meta fz ; object m ] - 1/3 * object ep end line
line ell big b end block

line ell x because 1rule deduction modus ponens ell big a indeed
for all object m indeed 0 < object ep and0 
  ( object n2 <= object m imply [ meta fx ; object m ] <= [ meta fz ; object m ] - object ep ) imply
0 < 1/3 * object ep end line

line ell y because pred lemma 2exist mp modus ponens ell x modus ponens ell g indeed 
  0 < 1/3 * object ep end line
  
line ell h because 1rule deduction modus ponens ell big b indeed
for all object m indeed ( 0 < 1/3 * object ep imply object n1 <= object m imply 
  | [ meta fx ; object m ] - [ meta fy ; object m ] | < 1/3 * object ep ) imply
for all object m indeed 0 < object ep and0 
  ( object n2 <= object m imply [ meta fx ; object m ] <= [ meta fz ; object m ] - object ep ) imply
max( object n1 , object n2 ) <= object m imply
[ meta fy ; object m ] <= [ meta fz ; object m ] - 1/3 * object ep end line

line ell i because pred lemma exist mp modus ponens ell h modus ponens ell e indeed
for all object m indeed 0 < object ep and0 
  ( object n2 <= object m imply [ meta fx ; object m ] <= [ meta fz ; object m ] - object ep ) imply
max( object n1 , object n2 ) <= object m imply
[ meta fy ; object m ] <= [ meta fz ; object m ] - 1/3 * object ep end line

line ell m because pred lemma 2exist mp modus ponens ell i modus ponens ell g indeed
  max( object n1 , object n2 ) <= object m imply
  [ meta fy ; object m ] <= [ meta fz ; object m ] - 1/3 * object ep end line

line ell z because prop lemma join conjuncts modus ponens ell y modus ponens ell m indeed
  0 < 1/3 * object ep and0 ( max( object n1 , object n2 ) <= object m imply
   [ meta fy ; object m ] <= [ meta fz ; object m ] - 1/3 * object ep ) end line

line ell n because 1rule gen modus ponens ell z indeed for all object m indeed 
  0 < 1/3 * object ep and0 ( max( object n1 , object n2 ) <= object m imply
  [ meta fy ; object m ] <= [ meta fz ; object m ] - 1/3 * object ep ) end line
line ell o because pred lemma intro exist at max( object n1 , object n2 ) modus ponens ell n indeed
  exist0 object n indeed for all object m indeed 0 < 1/3 * object ep and0 ( object n <= object m imply
  [ meta fy ; object m ] <= [ meta fz ; object m ] - 1/3 * object ep ) end line
line ell p because pred lemma intro exist at 1/3 * object ep modus ponens ell o indeed
  exist0 object ep indeed exist0 object n indeed for all object m indeed 0 < object ep and0 
  ( object n <= object m imply [ meta fy ; object m ] <= [ meta fz ; object m ] - object ep ) end line

because 1rule repetition modus ponens ell p indeed meta fy <f meta fz qed ""; XXREP <f
end math ]"

"[ math in theory system Q lemma lemma subLessLeft(R) says
for all terms meta fx comma meta fy comma meta fz indeed
R( meta fx ) = R( meta fy ) infer R( meta fx ) << R( meta fz ) infer R( meta fy ) << R( meta fz )
end lemma end math ]"

"[ math system Q proof of lemma subLessLeft(R) reads
any term meta fx comma meta fy comma meta fz end line
line ell a premise  R( meta fx ) = R( meta fy )  end line 
line ell b premise  R( meta fx ) << R( meta fz ) end line
line ell c because 1rule from== modus ponens ell a indeed meta fx sameF meta fy end line
line ell d because 1rule repetition modus ponens ell b indeed meta fx <f meta fz end line ""; XXREP <<
line ell e because lemma subLessLeft(F) modus ponens ell c modus ponens ell d indeed 
  meta fy <f meta fz end line
because 1rule repetition modus ponens ell e indeed R( meta fy ) << R( meta fz ) qed ""; XXREP <<
end math ]"




"[ math in theory system Q lemma lemma closetogreaterIsGreater says
for all terms meta x comma meta y1 comma meta y2 comma meta z indeed
meta x <= meta y1 - meta z infer | meta y1 - meta y2 | < 1/3 * meta z  infer
meta x <= meta y2 - 1/3 * meta z 
end lemma end math ]"

"[ math system Q proof of lemma closetogreaterIsGreater reads
any term meta x comma meta y1 comma meta y2 comma meta z end line
line ell a premise meta x <= meta y1 - meta z end line
line ell b premise | meta y1 - meta y2 | < 1/3 * meta z end line
line ell c because lemma 0<=|x| indeed 0 <= | meta y1 - meta y2 | end line
line ell d because lemma leqLessTransitivity modus ponens ell c modus ponens ell b indeed 
  0 < 1/3 * meta z end line

line ell e because lemma |x-x|=0 indeed | meta x - meta x | = 0 end line
line ell f because lemma eqSymmetry modus ponens ell e indeed 0 = | meta x - meta x | end line
line ell g because lemma subLessLeft modus ponens ell f modus ponens ell d indeed
  | meta x - meta x | < 1/3 * meta z end line

because lemma preserveLessGreater modus ponens ell a modus ponens ell g modus ponens ell b indeed 
meta x <= meta y2 - 1/3 * meta z qed
end math ]"


"[ math in theory system Q lemma lemma subLessRight(F) says
for all terms meta fx comma meta fy comma meta fz indeed
meta fx sameF meta fy infer meta fz <f meta fx infer meta fz <f meta fy
end lemma end math ]"

"[ math system Q proof of lemma subLessRight(F) reads
any term meta fx comma meta fy comma meta fz end line
line ell a premise meta fx sameF meta fy end line
line ell b premise meta fz <f meta fx end line


line ell c because 1rule repetition modus ponens ell a indeed ""; XXREP sameF PASPaa
  for all object ep indeed exist0 object n indeed for all object m indeed (
  0 < object ep imply object n <= object m imply 
  | [ meta fx ; object m ] - [ meta fy ; object m ] | < object ep ) end line

line ell d because 1rule deduction modus ponens ell c indeed ""; XXded PASPaa
  for all object ep indeed exist0 object n1 indeed for all object m indeed (
  0 < object ep imply object n1 <= object m imply 
  | [ meta fx ; object m ] - [ meta fy ; object m ] | < object ep ) end line

line ell e because lemma a4 at 1/3 * object ep modus ponens ell c indeed "";XX PASPaa
  exist0 object n1 indeed for all object m indeed (
  0 < 1/3 * object ep imply object n1 <= object m imply 
  | [ meta fx ; object m ] - [ meta fy ; object m ] | < 1/3 * object ep ) end line

line ell f because 1rule repetition modus ponens ell b indeed ""; XXREP <f
  exist0 object ep indeed exist0 object n indeed for all object m indeed 0 < object ep and0 
  ( object n <= object m imply [ meta fz ; object m ] <= [ meta fx ; object m ] - object ep ) end line

line ell g because 1rule deduction modus ponens ell f indeed ""; XXded
  exist0 object ep indeed exist0 object n2 indeed for all object m indeed 0 < object ep and0 
  ( object n2 <= object m imply [ meta fz ; object m ] <= [ meta fx ; object m ] - object ep ) end line

""; XX kan godt optimeres til 1 blok, men det kraever 
""; 1. 'intelligente' blokke 2. en aendring af <f-definitionen
block any term meta fx comma meta fx comma meta fz end line
line ell a premise for all object m indeed 0 < object ep and0 ""; XX linie g
  ( object n2 <= object m imply [ meta fz ; object m ] <= [ meta fx ; object m ] - object ep ) end line
line ell b because lemma a4 at object m modus ponens ell a indeed 0 < object ep and0 
  ( object n2 <= object m imply [ meta fz ; object m ] <= [ meta fx ; object m ] - object ep ) end line
line ell c because prop lemma first conjunct modus ponens ell b indeed 0 < object ep end line
because lemma positiveDividedBy3 modus ponens ell c indeed 0 < 1/3 * object ep end line
line ell big a end block


block any term meta fx comma meta fy comma meta fz end line
line ell a premise for all object m indeed ( ""; XX linie e PASPaa
  0 < 1/3 * object ep imply object n1 <= object m imply 
  | [ meta fx ; object m ] - [ meta fy ; object m ] | < 1/3 * object ep ) end line
line ell b premise for all object m indeed 0 < object ep and0  ""; XX linie g
  ( object n2 <= object m imply [ meta fz ; object m ] <= [ meta fx ; object m ] - object ep ) end line
line ell x premise max( object n1 , object n2 ) <= object m end line
line ell big b because lemma leqMax1 indeed object n1 <= max( object n1 , object n2 ) end line
line ell big c because lemma leqTransitivity modus ponens ell big b modus ponens ell x indeed
  object n1 <= object m end line
line ell big d because lemma leqMax2 indeed object n2 <= max( object n1 , object n2 ) end line
line ell big e because lemma leqTransitivity modus ponens ell big d modus ponens ell x indeed
  object n2 <= object m end line

line ell d because lemma a4 at object m modus ponens ell b indeed 0 < object ep and0
  ( object n2 <= object m imply [ meta fz ; object m ] <= [ meta fx ; object m ] - object ep ) end line
line ell e because prop lemma first conjunct modus ponens ell d indeed 0 < object ep end line
line ell f because prop lemma second conjunct modus ponens ell d indeed 
  object n2 <= object m imply [ meta fz ; object m ] <= [ meta fx ; object m ] - object ep end line
line ell g because 1rule mp modus ponens ell f modus ponens ell big e indeed
  [ meta fz ; object m ] <= [ meta fx ; object m ] - object ep end line

line ell h because lemma a4 at object m modus ponens ell a indeed ""; PASPaa
  0 < 1/3 * object ep imply object n1 <= object m imply 
  | [ meta fx ; object m ] - [ meta fy ; object m ] | < 1/3 * object ep end line
line ell i because lemma positiveDividedBy3 modus ponens ell e indeed 0 < 1/3 * object ep end line
line ell j because prop lemma mp2 modus ponens ell h modus ponens ell i modus ponens ell big c indeed
  | [ meta fx ; object m ] - [ meta fy ; object m ] | < 1/3 * object ep end line ""; PASPaa
because lemma closetogreaterIsGreater modus ponens ell g modus ponens ell j indeed
   [ meta fz ; object m ] <= [ meta fy ; object m ] - 1/3 * object ep end line
line ell big b end block

line ell x because 1rule deduction modus ponens ell big a indeed
for all object m indeed 0 < object ep and0 
  ( object n2 <= object m imply [ meta fz ; object m ] <= [ meta fx ; object m ] - object ep ) imply
0 < 1/3 * object ep end line

line ell y because pred lemma 2exist mp modus ponens ell x modus ponens ell g indeed 
  0 < 1/3 * object ep end line
  
line ell h because 1rule deduction modus ponens ell big b indeed
for all object m indeed ( 0 < 1/3 * object ep imply object n1 <= object m imply 
  | [ meta fx ; object m ] - [ meta fy ; object m ] | < 1/3 * object ep ) imply
for all object m indeed 0 < object ep and0 
  ( object n2 <= object m imply [ meta fz ; object m ] <= [ meta fx ; object m ] - object ep ) imply
max( object n1 , object n2 ) <= object m imply
[ meta fz ; object m ] <= [ meta fy ; object m ] - 1/3 * object ep end line

line ell i because pred lemma exist mp modus ponens ell h modus ponens ell e indeed
for all object m indeed 0 < object ep and0 
  ( object n2 <= object m imply [ meta fz ; object m ] <= [ meta fx ; object m ] - object ep ) imply
max( object n1 , object n2 ) <= object m imply
[ meta fz ; object m ] <= [ meta fy ; object m ] - 1/3 * object ep end line

line ell m because pred lemma 2exist mp modus ponens ell i modus ponens ell g indeed
  max( object n1 , object n2 ) <= object m imply
  [ meta fz ; object m ] <= [ meta fy ; object m ] - 1/3 * object ep end line

line ell z because prop lemma join conjuncts modus ponens ell y modus ponens ell m indeed
  0 < 1/3 * object ep and0 ( max( object n1 , object n2 ) <= object m imply
   [ meta fz ; object m ] <= [ meta fy ; object m ] - 1/3 * object ep ) end line

line ell n because 1rule gen modus ponens ell z indeed for all object m indeed 
  0 < 1/3 * object ep and0 ( max( object n1 , object n2 ) <= object m imply
  [ meta fz ; object m ] <= [ meta fy ; object m ] - 1/3 * object ep ) end line
line ell o because pred lemma intro exist at max( object n1 , object n2 ) modus ponens ell n indeed
  exist0 object n indeed for all object m indeed 0 < 1/3 * object ep and0 ( object n <= object m imply
  [ meta fz ; object m ] <= [ meta fy ; object m ] - 1/3 * object ep ) end line
line ell p because pred lemma intro exist at 1/3 * object ep modus ponens ell o indeed
  exist0 object ep indeed exist0 object n indeed for all object m indeed 0 < object ep and0 
  ( object n <= object m imply [ meta fz ; object m ] <= [ meta fy ; object m ] - object ep ) end line

because 1rule repetition modus ponens ell p indeed meta fz <f meta fy qed ""; XXREP <f
end math ]"


"[ math in theory system Q lemma lemma subLessRight(R) says
for all terms meta fx comma meta fy comma meta fz indeed
R( meta fx ) = R( meta fy ) infer R( meta fz ) << R( meta fx ) infer R( meta fz ) << R( meta fy )
end lemma end math ]"

"[ math system Q proof of lemma subLessRight(R) reads
any term meta fx comma meta fy comma meta fz end line
line ell a premise  R( meta fx ) = R( meta fy )  end line 
line ell b premise  R( meta fz ) << R( meta fx ) end line
line ell c because 1rule from== modus ponens ell a indeed meta fx sameF meta fy end line
line ell d because 1rule repetition modus ponens ell b indeed meta fz <f meta fx end line ""; XXREP <<
line ell e because lemma subLessRight(F) modus ponens ell c modus ponens ell d indeed 
  meta fz <f meta fy end line
because 1rule repetition modus ponens ell e indeed R( meta fz ) << R( meta fy ) qed ""; XXREP <<
end math ]"


"[  math in theory system Q lemma lemma -x*y=-(x*y) says
for all terms meta x comma meta y indeed
- meta x * meta y = - parenthesis meta x * meta y end parenthesis
end lemma end math ]"

"[ math system Q proof of lemma -x*y=-(x*y) reads
any term meta x comma meta y end line
line ell a because lemma times(-1)Left indeed (-1) * meta x = - meta x end line
line ell b because lemma eqMultiplication modus ponens ell a indeed 
  (-1) * meta x * meta y = - meta x * meta y end line
line ell c because lemma eqSymmetry modus ponens ell b indeed
  - meta x * meta y = (-1) * meta x * meta y end line

line ell d because axiom timesAssociativity indeed 
  (-1) * meta x * meta y = (-1) * parenthesis meta x * meta y end parenthesis end line

line ell e because lemma times(-1)Left indeed 
  (-1) * parenthesis meta x * meta y end parenthesis = 
   - parenthesis meta x * meta y end parenthesis end line 

because lemma eqTransitivity4 modus ponens ell c modus ponens ell d modus ponens ell e indeed 
  - meta x * meta y = - parenthesis meta x * meta y end parenthesis qed
end math ]"

"[  math in theory system Q lemma lemma leqMultiplicationLeft says
for all terms meta x comma meta y comma meta z indeed
0 <= meta z infer meta x <= meta y infer meta z * meta x <= meta z * meta y
end lemma end math ]"

"[ math system Q proof of lemma leqMultiplicationLeft reads
any term meta x comma meta y comma meta z end line
line ell a premise 0 <= meta z end line
line ell b premise meta x <= meta y end line
line ell c because lemma leqMultiplication modus ponens ell a modus ponens ell b indeed 
  meta x * meta z <= meta y * meta z end line
line ell d because axiom timesCommutativity indeed meta x * meta z = meta z * meta x end line
line ell e because lemma subLeqLeft modus ponens ell d modus ponens ell c indeed 
  meta z * meta x <= meta y * meta z end line

line ell f because axiom timesCommutativity indeed meta y * meta z = meta z * meta y end line
because lemma subLeqRight modus ponens ell f modus ponens ell e indeed 
 meta z * meta x <= meta z * meta y qed
end math ]"

"[  math in theory system Q lemma lemma sameFmultiplication helper says
for all terms meta v1 comma meta v2 comma meta m comma meta n comma meta ep comma meta fx comma meta fy comma meta fz indeed
for all meta m indeed parenthesis 
  0 < meta ep * 1/ meta v1 imply meta n <= meta m imply 
  | [ meta fx ; meta m ] - [ meta fy ; meta m ] | < meta ep * 1/ meta v1 end parenthesis imply

for all meta v2 indeed | [ meta fz ; meta v2 ] | < meta v1 imply

0 < meta ep imply meta n <= meta m imply
| [ meta fx *f meta fz ; meta m ] - [ meta fy *f meta fz ; meta m ] | < meta ep
end lemma end math ]"


"[ math system Q proof of lemma sameFmultiplication helper reads
""; XX meta n og meta v1 er 'eksistens-kvantificeret'
block any term meta v1 comma meta v2 comma meta m comma meta n comma meta ep comma meta fx comma meta fy 
comma meta fz end line 
line ell a premise for all meta m indeed parenthesis ""; rigtig praemis
  0 < meta ep * 1/ meta v1 imply meta n <= meta m imply 
  | [ meta fx ; meta m ] - [ meta fy ; meta m ] | < meta ep * 1/ meta v1 end parenthesis end line
line ell b premise for all meta v2 indeed | [ meta fz ; meta v2 ] | < meta v1 end line ""; fBounded 

line ell c premise 0 < meta ep end line
line ell d premise meta n <= meta m end line

line ell big f because lemma a4 at meta v2 modus ponens ell b indeed 
  | [ meta fz ; meta v2 ] | < meta v1 end line

line ell e because lemma 0<=|x| indeed 0 <= | [ meta fz ; meta v2 ] | end line
line ell f because lemma leqLessTransitivity modus ponens ell e modus ponens ell big f indeed 
  0 < meta v1 end line
line ell g because lemma positiveInverted modus ponens ell f indeed 0 < 1/ meta v1 end line

line ell h because lemma positiveFactors modus ponens ell c modus ponens ell g indeed
  0 < meta ep * 1/ meta v1 end line 

line ell big g because lemma a4 at meta m modus ponens ell a indeed
  0 < meta ep * 1/ meta v1 imply meta n <= meta m imply 
  | [ meta fx ; meta m ] - [ meta fy ; meta m ] | < meta ep * 1/ meta v1 end line

line ell i because prop lemma mp2 modus ponens ell big g modus ponens ell h modus ponens ell d indeed
  | [ meta fx ; meta m ] - [ meta fy ; meta m ] | < meta ep * 1/ meta v1 end line

line ell k because lemma reciprocalToLeft(Less) modus ponens ell f modus ponens ell i indeed 
  | [ meta fx ; meta m ] - [ meta fy ; meta m ] | * meta v1 < meta ep end line  

line ell l because lemma timesF indeed 
  [ meta fx *f meta fz ; meta m ] = [ meta fx ; meta m ] * [ meta fz ; meta m ] end line
line ell m because lemma timesF indeed 
  [ meta fy *f meta fz ; meta m ] = [ meta fy ; meta m ] * [ meta fz ; meta m ] end line
line ell n because lemma eqNegated modus ponens ell m indeed
  - [ meta fy *f meta fz ; meta m ] = 
  - parenthesis [ meta fy ; meta m ] * [ meta fz ; meta m ] end parenthesis end line

line ell big o because lemma -x*y=-(x*y) indeed - [ meta fy ; meta m ] * [ meta fz ; meta m ] =
  - parenthesis [ meta fy ; meta m ] * [ meta fz ; meta m ] end parenthesis end line

line ell big p because lemma eqSymmetry modus ponens ell big o indeed 
  - parenthesis [ meta fy ; meta m ] * [ meta fz ; meta m ] end parenthesis =
  - [ meta fy ; meta m ] * [ meta fz ; meta m ] end line

line ell big q because lemma eqTransitivity modus ponens ell n modus ponens ell big p indeed
  - [ meta fy *f meta fz ; meta m ] = - [ meta fy ; meta m ] * [ meta fz ; meta m ] end line

line ell big n  because lemma addEquations modus ponens ell l modus ponens ell big q indeed
  [ meta fx *f meta fz ; meta m ] - [ meta fy *f meta fz ; meta m ] = 
  [ meta fx ; meta m ] * [ meta fz ; meta m ] + 
  parenthesis - [ meta fy ; meta m ] end parenthesis * [ meta fz ; meta m ] end line

line ell o because lemma distributionOutLeft indeed
  [ meta fx ; meta m ] * [ meta fz ; meta m ] + 
  parenthesis - [ meta fy ; meta m ] end parenthesis * [ meta fz ; meta m ] =
  [ meta fz ; meta m ] * parenthesis [ meta fx ; meta m ] - [ meta fy ; meta m ] end parenthesis end line

line ell p because lemma eqTransitivity modus ponens ell big n modus ponens ell o indeed
  [ meta fx *f meta fz ; meta m ] - [ meta fy *f meta fz ; meta m ] =
  [ meta fz ; meta m ] * parenthesis [ meta fx ; meta m ] - [ meta fy ; meta m ] end parenthesis end line

line ell q because lemma sameNumerical modus ponens ell p indeed
  | [ meta fx *f meta fz ; meta m ] - [ meta fy *f meta fz ; meta m ] | =
  | [ meta fz ; meta m ] * parenthesis [ meta fx ; meta m ] - [ meta fy ; meta m ] end parenthesis | 
  end line

line ell r because lemma splitNumericalProduct indeed
  | [ meta fz ; meta m ] * parenthesis [ meta fx ; meta m ] - [ meta fy ; meta m ] end parenthesis | =
  | [ meta fz ; meta m ] | * | [ meta fx ; meta m ] - [ meta fy ; meta m ] | end line

line ell s because axiom timesCommutativity indeed
  | [ meta fz ; meta m ] | * | [ meta fx ; meta m ] - [ meta fy ; meta m ] | =
  | [ meta fx ; meta m ] - [ meta fy ; meta m ] | * | [ meta fz ; meta m ] | end line

line ell t because lemma eqTransitivity4 modus ponens ell q modus ponens ell r modus ponens ell s indeed
  | [ meta fx *f meta fz ; meta m ] - [ meta fy *f meta fz ; meta m ] | =
  | [ meta fx ; meta m ] - [ meta fy ; meta m ] | * | [ meta fz ; meta m ] | end line

line ell u because lemma eqSymmetry modus ponens ell t indeed
 | [ meta fx ; meta m ] - [ meta fy ; meta m ] | * | [ meta fz ; meta m ] | =
 | [ meta fx *f meta fz ; meta m ] - [ meta fy *f meta fz ; meta m ] | end line

line ell v because lemma a4 at [ meta fz ; meta m ] modus ponens ell b indeed
  | [ meta fz ; meta m ] | <= meta v1 end line

line ell big e because lemma 0<=|x| indeed 0 <= | [ meta fx ; meta m ] - [ meta fy ; meta m ] | end line

line ell x because lemma leqMultiplicationLeft modus ponens ell big e modus ponens ell v indeed
  | [ meta fx ; meta m ] - [ meta fy ; meta m ] | * | [ meta fz ; meta m ] | <=
  | [ meta fx ; meta m ] - [ meta fy ; meta m ] | * meta v1 end line

line ell y because lemma leqLessTransitivity modus ponens ell x modus ponens ell k indeed
  | [ meta fx ; meta m ] - [ meta fy ; meta m ] | * | [ meta fz ; meta m ] | < meta ep end line
  
because lemma subLessLeft modus ponens ell u modus ponens ell y indeed
 | [ meta fx *f meta fz ; meta m ] - [ meta fy *f meta fz ; meta m ] | < meta ep end line
line ell big a end block

any term  meta v1 comma meta v2 comma meta m comma meta n comma meta ep comma meta fx comma meta fy comma meta fz end line


because 1rule deduction modus ponens ell big a indeed
for all meta m indeed parenthesis 
  0 < meta ep * 1/ meta v1 imply meta n <= meta m imply 
  | [ meta fx ; meta m ] - [ meta fy ; meta m ] | < meta ep * 1/ meta v1 end parenthesis imply
for all meta v2 indeed | [ meta fz ; meta v2 ] | < meta v1 imply
0 < meta ep imply meta n <= meta m imply
| [ meta fx *f meta fz ; meta m ] - [ meta fy *f meta fz ; meta m ] | < meta ep qed
end math ]"



"[  math in theory system Q lemma lemma sameFmultiplication says
for all terms meta v1 comma meta v2 comma meta m comma meta n comma meta ep comma meta fx comma meta fy comma meta fz indeed
meta fx sameF meta fy infer meta fx *f meta fz sameF meta fy *f meta fz
end lemma end math ]"

"[ math system Q proof of lemma sameFmultiplication reads
any term meta v1 comma meta v2 comma meta m comma meta n comma meta ep comma meta fx comma meta fy comma meta fz end line
line ell a premise meta fx sameF meta fy end line
line ell x because 1rule repetition modus ponens ell a indeed ""; XXREP sameF
  for all object ep indeed exist0 object n indeed for all object m indeed ( 0 < object ep imply 
  object n <= object m imply | [ meta fx ; object m ] - [ meta fy ; object m ] | < object ep ) end line
line ell b because 1rule deduction modus ponens ell x indeed ""; XXded
  for all meta ep indeed exist0 meta n indeed for all meta m indeed parenthesis 
  0 < meta ep imply meta n <= meta m imply | [ meta fx ; meta m ] - [ meta fy ; meta m ] | < meta ep 
  end parenthesis end line
line ell c because lemma a4 at meta ep * 1/ meta v1 modus ponens ell b indeed
  exist0 meta n indeed for all meta m indeed parenthesis 
  0 < meta ep * 1/ meta v1 imply meta n <= meta m imply 
  | [ meta fx ; meta m ] - [ meta fy ; meta m ] | < meta ep * 1/ meta v1 end parenthesis end line

line ell d because lemma f-Bounded indeed
  exist0 meta v1 indeed for all meta v2 indeed | [ meta fz ; meta v2 ] | < meta v1 end line

line ell e because lemma sameFmultiplication helper indeed
for all meta m indeed parenthesis 
  0 < meta ep * 1/ meta v1 imply meta n <= meta m imply 
  | [ meta fx ; meta m ] - [ meta fy ; meta m ] | < meta ep * 1/ meta v1 end parenthesis imply
for all meta v2 indeed | [ meta fz ; meta v2 ] | < meta v1 imply
0 < meta ep imply meta n <= meta m imply
| [ meta fx *f meta fz ; meta m ] - [ meta fy *f meta fz ; meta m ] | < meta ep end line

line ell f because pred lemma exist mp2 modus ponens ell e modus ponens ell c modus ponens ell d indeed
  0 < meta ep imply meta n <= meta m imply
  | [ meta fx *f meta fz ; meta m ] - [ meta fy *f meta fz ; meta m ] | < meta ep end line

line ell g because 1rule gen modus ponens ell f indeed for all meta m indeed parenthesis  
  0 < meta ep imply meta n <= meta m imply
  | [ meta fx *f meta fz ; meta m ] - [ meta fy *f meta fz ; meta m ] | < meta ep end parenthesis end line

line ell h because pred lemma intro exist at meta n modus ponens ell g indeed 
exist0 meta n indeed for all meta m indeed parenthesis 0 < meta ep imply meta n <= meta m imply
  | [ meta fx *f meta fz ; meta m ] - [ meta fy *f meta fz ; meta m ] | < meta ep end parenthesis end line

line ell i because 1rule gen modus ponens ell h indeed for all meta ep indeed
  exist0 meta n indeed for all meta m indeed parenthesis 0 < meta ep imply meta n <= meta m imply
  | [ meta fx *f meta fz ; meta m ] - [ meta fy *f meta fz ; meta m ] | < meta ep end parenthesis end line

line ell y because 1rule deduction modus ponens ell i indeed for all object ep indeed ""; XXded
  exist0 object n indeed for all object m indeed ( 0 < object ep imply object n <= object m imply
  | [ meta fx *f meta fz ; object m ] - [ meta fy *f meta fz ; object m ] | < object ep ) end line

because 1rule repetition modus ponens ell y indeed 
  meta fx *f meta fz sameF meta fy *f meta fz qed ""; XXREP sameF
end math ]"



"[  math in theory system Q lemma lemma eqMultiplication(R) says
for all terms meta fx comma meta fy comma meta fz indeed
R( meta fx ) == R( meta fy ) infer R( meta fx ) ** R( meta fz ) == R( meta fy ) ** R( meta fz )
end lemma end math ]"


"[ math system Q proof of lemma eqMultiplication(R) reads
any term meta fx comma meta fy comma meta fz end line
line ell a premise R( meta fx ) == R( meta fy ) end line
line ell b because 1rule from== modus ponens ell a indeed meta fx sameF meta fy end line

line ell c because lemma sameFmultiplication modus ponens ell b indeed
  meta fx *f meta fz sameF meta fy *f meta fz end line
line ell d because 1rule to== modus ponens ell c indeed
  R( meta fx *f meta fz ) == R( meta fy *f meta fz ) end line

line ell e because lemma eqReflexivity indeed 
  R( meta fx ) ** R( meta fz ) = R( meta fx *f meta fz ) end line

line ell g because lemma eqReflexivity indeed
  R( meta fy *f meta fz ) == R( meta fy ) ** R( meta fz ) end line

"";line ell h because lemma ==Transitivity modus ponens ell e modus ponens ell d indeed
"";  R( meta fx ) ** R( meta fz ) == R( meta fy *f meta fz ) end line
because lemma eqTransitivity4 modus ponens ell e modus ponens ell d modus ponens ell g indeed
  R( meta fx ) ** R( meta fz ) = R( meta fy ) ** R( meta fz ) qed
end math ]"

"[  math in theory system Q lemma lemma x*0=0(F) says
for all terms meta m comma meta fx indeed ""; XX lidt grimt med den ekstra variabel
meta fx *f 0f = 0f
end lemma end math ]"

"[ math system Q proof of lemma x*0=0(F) reads
any term meta m comma meta fx end line
line ell big a because lemma timesF indeed 
  [ meta fx *f 0f ; meta m ] = [ meta fx ; meta m ] * [ 0f ; meta m ] end line

line ell x because axiom natType indeed meta m in0 N end line
line ell a because lemma 0f modus ponens ell x indeed [ 0f ; meta m ] = 0 end line
line ell b because lemma eqMultiplicationLeft modus ponens ell a indeed
  [ meta fx ; meta m ] * [ 0f ; meta m ] = [ meta fx ; meta m ] * 0 end line

line ell c because lemma x*0=0 indeed [ meta fx ; meta m ] * 0 = 0 end line

line ell d because lemma eqSymmetry modus ponens ell a indeed 0 = [ 0f ; meta m ] end line

line ell e because lemma eqTransitivity5 modus ponens ell big a modus ponens ell b 
  modus ponens ell c modus ponens ell d indeed [ meta fx *f 0f ; meta m ] = [ 0f ; meta m ] end line

line ell f because 1rule gen modus ponens ell e indeed
  for all meta m indeed [ meta fx *f 0f ; meta m ] = [ 0f ; meta m ] end line

because lemma to=f modus ponens ell f indeed meta fx *f 0f = 0f qed
end math ]"

"[  math in theory system Q lemma lemma x*0=0(R) says
for all terms meta fx indeed
R( meta fx ) ** 00 == 00 
end lemma end math ]"

"[ math system Q proof of lemma x*0=0(R) reads
any term meta fx end line
line ell a because lemma x*0=0(F) indeed meta fx *f 0f = 0f end line
line ell b because lemma =f to sameF modus ponens ell a indeed meta fx *f 0f sameF 0f end line
line ell c because 1rule to== modus ponens ell b indeed R( meta fx *f 0f ) == R( 0f ) end line
line ell d because lemma eqReflexivity indeed R( meta fx ) ** R( 0f ) == R( meta fx *f 0f ) end line
line ell e because lemma ==Transitivity modus ponens ell d modus ponens ell c indeed
  R( meta fx ) ** R( 0f ) == R( 0f ) end line
because 1rule repetition modus ponens ell e indeed R( meta fx ) ** 00 == 00 qed ""; XXREP 00
end math ]"



"[  math in theory system Q lemma lemma distributionLeft says
for all terms meta x comma meta y comma meta z indeed
meta x * parenthesis meta y + meta z end parenthesis = 
meta y * meta x + meta z * meta x 
end lemma end math ]"

"[ math system Q proof of lemma distributionLeft reads
any term meta x comma meta y comma meta z end line
line ell a because lemma distributionOutLeft indeed
  meta y * meta x + meta z * meta x = meta x * parenthesis meta y + meta z end parenthesis end line
because lemma eqSymmetry modus ponens ell a indeed 
  meta x * parenthesis meta y + meta z end parenthesis = meta y * meta x + meta z * meta x qed
end math ]"

XX 'nonnegativeFactors' er en konsekvens heraf
"[  math in theory system Q lemma lemma multiplyEquations(Leq) says
for all terms meta x comma meta y comma meta z comma meta u indeed
0 <= meta x infer 0 <= meta z infer 
meta x <= meta y infer meta z <= meta u infer meta x * meta z <= meta y * meta u 
end lemma end math ]"

"[ math system Q proof of lemma multiplyEquations(Leq) reads
any term meta x comma meta y comma meta z comma meta u end line
line ell big x premise 0 <= meta x end line
line ell big z premise 0 <= meta z end line
line ell a premise meta x <= meta y end line
line ell b premise meta z <= meta u end line
line ell c because lemma leqMultiplication modus ponens ell big z modus ponens ell a indeed 
  meta x * meta z <= meta y * meta z end line

line ell big y because lemma leqTransitivity modus ponens ell big x modus ponens ell a indeed
  0 <= meta y end line

line ell d because lemma leqMultiplicationLeft modus ponens ell big y modus ponens ell b indeed
  meta y * meta z <= meta y * meta u end line
because lemma leqTransitivity modus ponens ell c modus ponens ell d indeed 
  meta x * meta z <= meta y * meta u qed
end math ]"


"[  math in theory system Q lemma lemma lessMultiplication(F) helper2 says
for all terms meta x comma meta y comma meta z comma meta u comma meta v indeed
0 < meta u infer 0 < meta v infer
meta x <= meta y - meta u infer
0 <= meta z - meta v infer
meta x * meta z <= meta y * meta z - parenthesis meta u * meta v end parenthesis 
end lemma end math ]"

"[ math system Q proof of lemma lessMultiplication(F) helper2 reads
any term meta x comma meta y comma meta z comma meta u comma meta v end line
line ell big j premise 0 < meta u end line
line ell big n premise 0 < meta v end line
line ell big a premise meta x <= meta y - meta u end line
line ell big b premise 0 <= meta z - meta v end line

line ell big c because lemma negativeToLeft(Leq) modus ponens ell big a indeed
  meta x + meta u <= meta y end line
line ell big d because axiom plusCommutativity indeed meta x + meta u = meta u + meta x end line
line ell big e because lemma subLeqLeft modus ponens ell big d modus ponens ell big c indeed
  meta u + meta x <= meta y end line

line ell a because lemma positiveToRight(Leq) modus ponens ell big e indeed 
  meta u <= meta y - meta x end line
line ell b because lemma negativeToLeft(Leq)(1 term) modus ponens ell big b indeed meta v <= meta z end line

line ell big k because lemma lessLeq modus ponens ell big j indeed 0 <= meta u end line
line ell big o because lemma lessLeq modus ponens ell big n indeed 0 <= meta v end line

line ell c because lemma multiplyEquations(Leq) modus ponens ell big k modus ponens ell big o 
  modus ponens ell a modus ponens ell b indeed 
  meta u * meta v <= parenthesis meta y - meta x end parenthesis * meta z end line

line ell d because axiom timesCommutativity indeed 
  parenthesis meta y - meta x end parenthesis * meta z = 
  meta z * parenthesis meta y - meta x end parenthesis end line
line ell e because lemma distributionLeft indeed
  meta z * parenthesis meta y - meta x end parenthesis = 
  meta y * meta z + parenthesis - meta x end parenthesis * meta z end line

line ell big g because lemma -x*y=-(x*y) indeed - meta x * meta z = - parenthesis meta x * meta z end parenthesis end line
line ell big h because lemma eqAdditionLeft modus ponens ell big g indeed
  meta y * meta z + parenthesis - meta x end parenthesis * meta z = 
  meta y * meta z - parenthesis meta x * meta z end parenthesis end line

line ell f because lemma eqTransitivity4 modus ponens ell d modus ponens ell e modus ponens ell big h 
   indeed
  parenthesis meta y - meta x end parenthesis * meta z = 
  meta y * meta z - parenthesis meta x * meta z end parenthesis end line

line ell g because lemma subLeqRight modus ponens ell f modus ponens ell c indeed
  meta u * meta v <= meta y * meta z - parenthesis meta x * meta z end parenthesis end line

line ell h because lemma negativeToLeft(Leq) modus ponens ell g indeed 
  meta u * meta v + meta x * meta z <= meta y * meta z end line

line ell i because axiom plusCommutativity indeed 
  meta u * meta v + meta x  * meta z = 
  meta x * meta z + meta u * meta v end line

line ell j because lemma subLeqLeft modus ponens ell i modus ponens ell h indeed
  meta x * meta z + meta u * meta v <= meta y * meta z end line

because lemma positiveToRight(Leq) modus ponens ell j indeed 
  meta x * meta z <= meta y * meta z - parenthesis meta u * meta v end parenthesis qed
end math ]"

"[  math in theory system Q lemma lemma lessMultiplication(F) helper says
  for all terms meta m comma meta n comma meta ep1 comma meta ep2 comma meta fx comma meta fy comma meta fz 
  indeed 
  for all meta m indeed 0 < meta ep1 and0
  parenthesis meta n <= meta m imply [ meta fx ; meta m ] <= [ meta fy ; meta m ] - meta ep1 end parenthesis
  imply
  for all meta m indeed 0 < meta ep2 and0
  parenthesis meta n <= meta m imply [ 0f ; meta m ] <= [ meta fz ; meta m ] - meta ep2 end parenthesis
  imply 
  0 < meta ep1 * meta ep2 and0 parenthesis meta n <= meta m imply
  [ meta fx *f meta fz ; meta m ] <= 
  [ meta fy *f meta fz ; meta m ] - parenthesis meta ep1 * meta ep2 end parenthesis end parenthesis 
end lemma end math ]"

"[ math system Q proof of lemma lessMultiplication(F) helper reads

block any term meta m comma meta n comma meta ep1 comma meta ep2 comma meta fx comma meta fy comma meta fz 
end line ""; epsilon-blokken
line ell a premise for all meta m indeed 0 < meta ep1 and0 
  parenthesis meta n <= meta m imply [ meta fx ; meta m ] <= [ meta fy ; meta m ] - meta ep1 end parenthesis
  end line
line ell b premise for all meta m indeed 0 < meta ep2 and0
  parenthesis meta n <= meta m imply [ 0f ; meta m ] <= [ meta fz ; meta m ] - meta ep2 end parenthesis
  end line
line ell big b because lemma a4 at meta m modus ponens ell a indeed 0 < meta ep1 and0 
  parenthesis meta n <= meta m imply [ meta fx ; meta m ] <= [ meta fy ; meta m ] - meta ep1 end parenthesis
  end line
line ell big c because lemma a4 at meta m modus ponens ell b indeed 0 < meta ep2 and0
  parenthesis meta n <= meta m imply [ 0f ; meta m ] <= [ meta fz ; meta m ] - meta ep2 end parenthesis
  end line

line ell d because prop lemma first conjunct modus ponens ell big b indeed 0 < meta ep1 end line
line ell e because prop lemma first conjunct modus ponens ell big c indeed 0 < meta ep2 end line
because lemma positiveFactors modus ponens ell d modus ponens ell e indeed
  0 < meta ep1 * meta ep2 end line
line ell big a end block


block any term meta m comma meta n comma meta ep1 comma meta ep2 comma meta fx comma meta fy comma meta fz 
end line ""; implikations-blokken

line ell a premise for all meta m indeed 0 < meta ep1 and0
  parenthesis meta n <= meta m imply [ meta fx ; meta m ] <= [ meta fy ; meta m ] - meta ep1 end parenthesis
  end line
line ell b premise for all meta m indeed 0 < meta ep2 and0
  parenthesis meta n <= meta m imply [ 0f ; meta m ] <= [ meta fz ; meta m ] - meta ep2 end parenthesis
  end line
line ell c premise meta n <= meta m end line

line ell big b because lemma a4 at meta m modus ponens ell a indeed 0 < meta ep1 and0 
  parenthesis meta n <= meta m imply [ meta fx ; meta m ] <= [ meta fy ; meta m ] - meta ep1 end parenthesis
  end line
line ell big c because lemma a4 at meta m modus ponens ell b indeed 0 < meta ep2 and0
  parenthesis meta n <= meta m imply [ 0f ; meta m ] <= [ meta fz ; meta m ] - meta ep2 end parenthesis
  end line

line ell d because prop lemma first conjunct modus ponens ell big b indeed 0 < meta ep1 end line
line ell e because prop lemma first conjunct modus ponens ell big c indeed 0 < meta ep2 end line

line ell g because prop lemma second conjunct modus ponens ell big b indeed 
  meta n <= meta m imply [ meta fx ; meta m ] <= [ meta fy ; meta m ] - meta ep1 end line
line ell h because 1rule mp modus ponens ell g modus ponens ell c indeed
  [ meta fx ; meta m ] <= [ meta fy ; meta m ] - meta ep1 end line


line ell j because prop lemma second conjunct modus ponens ell big c indeed 
  meta n <= meta m imply [ 0f ; meta m ] <= [ meta fz ; meta m ] - meta ep2 end line
line ell k because 1rule mp modus ponens ell j modus ponens ell c indeed
  [ 0f ; meta m ] <= [ meta fz ; meta m ] - meta ep2 end line
line ell x because axiom natType indeed meta m in0 N end line
line ell l  because lemma 0f modus ponens ell x indeed [ 0f ; meta m ] = 0 end line
line ell m because lemma subLeqLeft modus ponens ell l modus ponens ell k indeed
  0 <= [ meta fz ; meta m ] - meta ep2 end line

line ell n because lemma lessMultiplication(F) helper2 
  modus ponens ell d modus ponens ell e modus ponens ell h modus ponens ell m indeed
  [ meta fx ; meta m ] * [ meta fz ; meta m ] <= 
  [ meta fy ; meta m ] * [ meta fz ; meta m ] - parenthesis meta ep1 * meta ep2 end parenthesis end line
  
line ell o because lemma timesF(Sym) indeed 
  [ meta fx ; meta m ] * [ meta fz ; meta m ] = [ meta fx *f meta fz ; meta m ] end line
line ell p because lemma subLeqLeft modus ponens ell o modus ponens ell n indeed
  [ meta fx *f meta fz ; meta m ] <= 
  [ meta fy ; meta m ] * [ meta fz ; meta m ] - parenthesis meta ep1 * meta ep2 end parenthesis end line

line ell q because lemma timesF(Sym) indeed 
  [ meta fy ; meta m ] * [ meta fz ; meta m ] = [ meta fy *f meta fz ; meta m ] end line
line ell r because lemma eqAddition modus ponens ell q indeed
  [ meta fy ; meta m ] * [ meta fz ; meta m ] - parenthesis meta ep1 * meta ep2 end parenthesis = 
  [ meta fy *f meta fz ; meta m ] - parenthesis meta ep1 * meta ep2 end parenthesis end line

because lemma subLeqRight modus ponens ell r modus ponens ell p indeed [ meta fx *f meta fz ; meta m ] <= 
  [ meta fy *f meta fz ; meta m ] - parenthesis meta ep1 * meta ep2 end parenthesis end line
line ell big b end block

any term meta m comma meta n comma meta ep1 comma meta ep2 comma meta fx comma meta fy comma meta fz 
end line

line ell a because 1rule deduction modus ponens ell big a indeed for all meta m indeed 0 < meta ep1 and0
  parenthesis meta n <= meta m imply [ meta fx ; meta m ] <= [ meta fy ; meta m ] - meta ep1 end parenthesis
  imply
  for all meta m indeed 0 < meta ep2 and0
  parenthesis meta n <= meta m imply [ 0f ; meta m ] <= [ meta fz ; meta m ] - meta ep2 end parenthesis
  imply 
  0 < meta ep1 * meta ep2 end line

line ell b because 1rule deduction modus ponens ell big b indeed for all meta m indeed 0 < meta ep1 and0
  parenthesis meta n <= meta m imply [ meta fx ; meta m ] <= [ meta fy ; meta m ] - meta ep1 end parenthesis
  imply
  for all meta m indeed 0 < meta ep2 and0 
  parenthesis meta n <= meta m imply [ 0f ; meta m ] <= [ meta fz ; meta m ] - meta ep2 end parenthesis
  imply 
  meta n <= meta m imply
  [ meta fx *f meta fz ; meta m ] <= [ meta fy *f meta fz ; meta m ] - parenthesis meta ep1 * meta ep2 end parenthesis end line

because prop lemma doubly conditioned join conjuncts modus ponens ell a modus ponens ell b indeed
  for all meta m indeed 0 < meta ep1 and0 parenthesis meta n <= meta m imply 
  [ meta fx ; meta m ] <= [ meta fy ; meta m ] - meta ep1 end parenthesis
  imply
  for all meta m indeed 0 < meta ep2 and0
  parenthesis meta n <= meta m imply [ 0f ; meta m ] <= [ meta fz ; meta m ] - meta ep2 end parenthesis
  imply 0 < meta ep1 * meta ep2 and0 parenthesis
  meta n <= meta m imply
  [ meta fx *f meta fz ; meta m ] <= [ meta fy *f meta fz ; meta m ] - parenthesis meta ep1 * meta ep2 
  end parenthesis end parenthesis qed
end math ]"

"[  math in theory system Q lemma lemma lessMultiplication(F) says 
""; XX lidt grimt med de ekstra variable
for all terms meta m comma meta n comma meta ep1 comma meta ep2
                     comma meta fx comma meta fy comma meta fz indeed
  0f <f meta fz infer meta fx <f meta fy infer meta fx *f meta fz <f meta fy *f meta fz 
end lemma end math ]"

"[ math system Q proof of lemma lessMultiplication(F) reads
any term meta m comma meta n comma meta ep1 comma meta ep2
                comma meta fx comma meta fy comma meta fz end line
line ell a premise 0f <f meta fz end line
line ell b premise meta fx <f meta fy end line
line ell x because 1rule repetition modus ponens ell b indeed ""; XXREP <f
  exist0 object ep indeed exist0 object n indeed for all object m indeed 0 < object ep and0 
  ( object n <= object m imply [ meta fx ; object m ] <= [ meta fy ; object m ] - object ep ) end line
line ell c because 1rule deduction modus ponens ell x indeed ""; XX instantiering af 'ep', XXded
  exist0 meta ep1 indeed exist0 meta n indeed for all meta m indeed 0 < meta ep1 and0 
  parenthesis meta n <= meta m imply [ meta fx ; meta m ] <= [ meta fy ; meta m ] - meta ep1 end parenthesis
  end line

line ell y because 1rule repetition modus ponens ell a indeed ""; XXREP <f
  exist0 object ep indeed exist0 object n indeed for all object m indeed 0 < object ep and0 
  ( object n <= object m imply [ 0f ; object m ] <= [ meta fz ; object m ] - object ep ) end line
line ell d because 1rule deduction modus ponens ell y indeed ""; XX instantiering af 'ep', XXded
  exist0 meta ep2 indeed exist0 meta n indeed for all meta m indeed 0 < meta ep2 and0 
  ( meta n <= meta m imply [ 0f ; meta m ] <= [ meta fz ; meta m ] - meta ep2 ) end line

line ell e because lemma lessMultiplication(F) helper indeed
  for all meta m indeed 0 < meta ep1 and0 
  parenthesis meta n <= meta m imply [ meta fx ; meta m ] <= [ meta fy ; meta m ] - meta ep1 end parenthesis
  imply
  for all meta m indeed 0 < meta ep2 and0 
  parenthesis meta n <= meta m imply [ 0f ; meta m ] <= [ meta fz ; meta m ] - meta ep2 end parenthesis
  imply 
  0 < meta ep1 * meta ep2 and0 parenthesis meta n <= meta m imply 
  [ meta fx *f meta fz ; meta m ] <= [ meta fy *f meta fz ; meta m ] 
  - parenthesis meta ep1 * meta ep2 end parenthesis end parenthesis end line

line ell f because pred lemma 2exist mp2 modus ponens ell e modus ponens ell c modus ponens ell d indeed
  0 < meta ep1 * meta ep2 and0 parenthesis meta n <= meta m imply 
  [ meta fx *f meta fz ; meta m ] <= [ meta fy *f meta fz ; meta m ] - 
  parenthesis meta ep1 * meta ep2 end parenthesis end parenthesis end line

line ell i because 1rule gen modus ponens ell f indeed for all meta m indeed 0 < meta ep1 * meta ep2 and0  
  parenthesis meta n <= meta m imply [ meta fx *f meta fz ; meta m ] <= [ meta fy *f meta fz ; meta m ] - 
  parenthesis meta ep1 * meta ep2 end parenthesis end parenthesis end line

line ell j because pred lemma intro exist at meta n modus ponens ell i indeed 
  exist0 meta n indeed for all meta m indeed 0 < meta ep1 * meta ep2 and0  
  parenthesis meta n <= meta m imply [ meta fx *f meta fz ; meta m ] <= [ meta fy *f meta fz ; meta m ] - 
  parenthesis meta ep1 * meta ep2 end parenthesis end parenthesis end line

line ell l because pred lemma intro exist at meta ep1 * meta ep2 modus ponens ell j indeed 
  exist0 meta ep indeed exist0 meta n indeed for all meta m indeed 0 < meta ep and0 parenthesis  
  meta n <= meta m imply [ meta fx *f meta fz ; meta m ] <= [ meta fy *f meta fz ; meta m ] - meta ep
  end parenthesis end line

line ell m because 1rule deduction modus ponens ell l indeed ""; XXded 
  exist0 object ep indeed exist0 object n indeed for all object m indeed 0 < object ep and0 
  ( object n <= object m imply 
  [ meta fx *f meta fz ; object m ] <= [ meta fy *f meta fz ; object m ] - object ep ) end line


because 1rule repetition modus ponens ell m indeed meta fx *f meta fz <f meta fy *f meta fz qed ""; XXREP <f 
end math ]"



"[  math in theory system Q lemma lemma lessMultiplication(R) says
for all terms meta fx comma meta fy comma meta fz indeed
00 << R( meta fz ) infer R( meta fx ) << R( meta fy ) infer
R( meta fx ) ** R( meta fz ) << R( meta fy ) ** R( meta fz )
end lemma end math ]"

"[ math system Q proof of lemma lessMultiplication(R) reads
any term meta fx comma meta fy comma meta fz end line
line ell a premise 00 << R( meta fz ) end line

line ell b premise R( meta fx ) << R( meta fy ) end line
line ell c because 1rule repetition modus ponens ell a indeed 0f <f meta fz end line  ""; XXREP <<
line ell d because 1rule repetition modus ponens ell b indeed meta fx <f meta fy end line ""; XXREP <<
line ell e because lemma lessMultiplication(F) modus ponens ell c modus ponens ell d indeed
  meta fx *f meta fz <f meta fy *f meta fz end line
line ell f because 1rule repetition modus ponens ell e indeed ""; XXREP <<
  R( meta fx *f meta fz ) << R( meta fy *f meta fz ) end line

line ell g because lemma eqReflexivity indeed 
  R( meta fx *f meta fz ) = R( meta fx ) ** R( meta fz ) end line
line ell h because lemma subLessLeft(R) modus ponens ell g modus ponens ell f indeed
  R( meta fx ) ** R( meta fz ) << R( meta fy *f meta fz ) end line

line ell i because lemma eqReflexivity indeed 
  R( meta fy *f meta fz ) = R( meta fy ) ** R( meta fz ) end line
because lemma subLessRight(R) modus ponens ell i modus ponens ell h indeed 
  R( meta fx ) ** R( meta fz ) << R( meta fy ) ** R( meta fz ) qed
end math ]"

---------------

"[  math in theory system Q lemma lemma eqMultiplicationLeft(R) says
for all terms meta fx comma meta fy comma meta fz indeed
R( meta fx ) == R( meta fy ) infer R( meta fz ) ** R( meta fx ) == R( meta fz ) ** R( meta fy )
end lemma end math ]"

"[ math system Q proof of lemma eqMultiplicationLeft(R) reads
any term meta fx comma meta fy comma meta fz end line
line ell a premise R( meta fx ) == R( meta fy ) end line
line ell b because lemma eqMultiplication(R) modus ponens ell a indeed 
  R( meta fx ) ** R( meta fz ) == R( meta fy ) ** R( meta fz ) end line
line ell c because lemma timesCommutativity(R) indeed 
  R( meta fz ) ** R( meta fx ) == R( meta fx ) ** R( meta fz ) end line
line ell d because lemma timesCommutativity(R) indeed 
  R( meta fy ) ** R( meta fz ) == R( meta fz ) ** R( meta fy ) end line

line ell e because lemma ==Transitivity modus ponens ell c modus ponens ell b indeed
  R( meta fz ) ** R( meta fx ) == R( meta fy ) ** R( meta fz ) end line
 because lemma ==Transitivity modus ponens ell e modus ponens ell d indeed
  R( meta fz ) ** R( meta fx ) == R( meta fz ) ** R( meta fy ) qed
end math ]"

"[  math in theory system Q lemma lemma leqMultiplication(R) says
for all terms meta fx comma meta fy comma meta fz indeed
00 <<== R( meta fz ) infer R( meta fx ) <<== R( meta fy ) infer 
R( meta fx ) ** R( meta fz ) <<== R( meta fy ) ** R( meta fz )
end lemma end math ]"

"[ math system Q proof of lemma leqMultiplication(R) reads
 

block any term meta fx comma meta fy comma meta fz end line
line ell a premise 00 == R( meta fz ) end line
line ell b because lemma ==Symmetry modus ponens ell a indeed R( meta fz ) == 00 end line
line ell c because lemma eqMultiplicationLeft(R) modus ponens ell b indeed
  R( meta fx ) ** R( meta fz ) == R( meta fx ) ** 00 end line

line ell d because lemma x*0=0(R) indeed R( meta fx ) ** 00 == 00 end line

line ell e because lemma x*0=0(R) indeed R( meta fy ) ** 00 == 00 end line
line ell f because lemma ==Symmetry modus ponens ell e indeed 00 == R( meta fy ) ** 00 end line

line ell g because lemma eqMultiplicationLeft(R) modus ponens ell a indeed
  R( meta fy ) ** 00 == R( meta fy ) ** R( meta fz ) end line

line ell h because lemma ==Transitivity modus ponens ell c modus ponens ell d indeed
  R( meta fx ) ** R( meta fz ) == 00 end line
line ell i because lemma ==Transitivity modus ponens ell h modus ponens ell f indeed
  R( meta fx ) ** R( meta fz ) == R( meta fy ) ** 00 end line
line ell j because lemma ==Transitivity modus ponens ell i modus ponens ell g indeed
  R( meta fx ) ** R( meta fz ) == R( meta fy ) ** R( meta fz ) end line

because lemma eqLeq(R) modus ponens ell j indeed 
  R( meta fx ) ** R( meta fz ) <<== R( meta fy ) ** R( meta fz ) end line
line ell big a end block

block any term meta fx comma meta fy comma meta fz end line
line ell a premise R( meta fx ) == R( meta fy ) end line
line ell b because lemma eqMultiplication(R) modus ponens ell a indeed 
  R( meta fx ) ** R( meta fz ) == R( meta fy ) ** R( meta fz ) end line
because lemma eqLeq(R) modus ponens ell b indeed 
  R( meta fx ) ** R( meta fz ) <<== R( meta fy ) ** R( meta fz ) end line
line ell big b end block

block any term meta fx comma meta fy comma meta fz end line
line ell a premise 00 << R( meta fz ) and0 R( meta fx ) << R( meta fy ) end line
line ell b because prop lemma first conjunct modus ponens ell a indeed 00 << R( meta fz ) end line
line ell c because prop lemma second conjunct modus ponens ell a indeed 
  R( meta fx ) << R( meta fy ) end line
line ell d because lemma lessMultiplication(R) modus ponens ell b modus ponens ell c indeed
  R( meta fx ) ** R( meta fz ) << R( meta fy ) ** R( meta fz ) end line
because lemma lessLeq(R) modus ponens ell d indeed
  R( meta fx ) ** R( meta fz ) <<== R( meta fy ) ** R( meta fz ) end line
line ell big c end block


any term meta fx comma meta fy comma meta fz end line

line ell a because 1rule deduction modus ponens ell big a indeed 00 == R( meta fz ) imply
  R( meta fx ) ** R( meta fz ) <<== R( meta fy ) ** R( meta fz ) end line

line ell b because 1rule deduction modus ponens ell big b indeed R( meta fx ) == R( meta fy ) imply 
  R( meta fx ) ** R( meta fz ) <<== R( meta fy ) ** R( meta fz ) end line

line ell c because 1rule deduction modus ponens ell big c indeed 
  00 << R( meta fz ) and0 R( meta fx ) << R( meta fy ) imply
  R( meta fx ) ** R( meta fz ) <<== R( meta fy ) ** R( meta fz ) end line

line ell d premise 00 <<== R( meta fz ) end line
line ell e premise R( meta fx ) <<== R( meta fy ) end line

line ell f because 1rule repetition modus ponens ell d indeed ""; XXREP <<==
  00 << R( meta fz ) or0 00 == R( meta fz ) end line

line ell g because 1rule repetition modus ponens ell e indeed ""; XXREP <<==
  R( meta fx ) << R( meta fy ) or0 R( meta fx ) == R( meta fy ) end line

line ell h because prop lemma expand disjuncts modus ponens ell f modus ponens ell g indeed
  00 == R( meta fz ) or0 R( meta fx ) == R( meta fy ) or0 
  parenthesis 00 << R( meta fz ) and0 R( meta fx ) << R( meta fy ) end parenthesis end line

because prop lemma from three disjuncts modus ponens ell h modus ponens ell a modus ponens ell b
  modus ponens ell c indeed R( meta fx ) ** R( meta fz ) <<== R( meta fy ) ** R( meta fz ) qed
end math ]"

"[  math in theory system Q lemma lemma 2cauchy helper says
for all terms meta v1 comma meta v2 comma meta n1 comma meta n2 comma meta ep 
                      comma meta fx comma meta fy indeed
for all meta v1 comma meta v2 indeed parenthesis 0 < meta ep imply 
  meta n1 <= meta v1 imply meta n1 <= meta v2 imply
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < meta ep end parenthesis imply
for all meta v1 comma meta v2 indeed parenthesis 0 < meta ep imply 
  meta n2 <= meta v1 imply meta n2 <= meta v2 imply
  | [ meta fy ; meta v1 ] - [ meta fy ; meta v2 ] | < meta ep end parenthesis imply
0 < meta ep imply max( meta n1 , meta n2 ) <= meta v1 imply max( meta n1 , meta n2 ) <= meta v2 imply
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < meta ep and0
  | [ meta fy ; meta v1 ] - [ meta fy ; meta v2 ] | < meta ep
end lemma end math ]"

"[ math system Q proof of lemma 2cauchy helper reads

block any term meta v1 comma meta v2 comma meta n1 comma meta n2 comma meta ep 
                       comma meta fx comma meta fy end line

line ell a premise for all meta v1 comma meta v2 indeed parenthesis 0 < meta ep imply 
  meta n1 <= meta v1 imply meta n1 <= meta v2 imply
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < meta ep end parenthesis end line
line ell b premise for all meta v1 comma meta v2 indeed parenthesis 0 < meta ep imply 
  meta n2 <= meta v1 imply meta n2 <= meta v2 imply
  | [ meta fy ; meta v1 ] - [ meta fy ; meta v2 ] | < meta ep end parenthesis end line

line ell c premise 0 < meta ep end line
line ell d premise max( meta n1 , meta n2 ) <= meta v1 end line
line ell e premise max( meta n1 , meta n2 ) <= meta v2 end line

line ell f because lemma leqMax1 indeed meta n1 <= max( meta n1 , meta n2 ) end line
line ell g because lemma leqTransitivity modus ponens ell f modus ponens ell d indeed 
  meta n1 <= meta v1 end line
line ell h because lemma leqTransitivity modus ponens ell f modus ponens ell e indeed
  meta n1 <= meta v2 end line

line ell big a because lemma leqMax2 indeed meta n2 <= max( meta n1 , meta n2 ) end line
line ell big b because lemma leqTransitivity modus ponens ell big a modus ponens ell d indeed
  meta n2 <= meta v1 end line
line ell i because lemma leqTransitivity modus ponens ell big a modus ponens ell e indeed
  meta n2 <= meta v2 end line

line ell j because lemma a4 at meta v1 modus ponens ell a indeed
  for all meta v2 indeed parenthesis 0 < meta ep imply 
  meta n1 <= meta v1 imply meta n1 <= meta v2 imply
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < meta ep end parenthesis end line
line ell k because lemma a4 at meta v2 modus ponens ell j indeed
  0 < meta ep imply meta n1 <= meta v1 imply meta n1 <= meta v2 imply
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < meta ep end line
line ell l because prop lemma mp3 modus ponens ell k modus ponens ell c modus ponens ell g 
  modus ponens ell h indeed | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < meta ep end line

line ell m because lemma a4 at meta v1 modus ponens ell b indeed
  for all meta v2 indeed parenthesis 0 < meta ep imply 
  meta n2 <= meta v1 imply meta n2 <= meta v2 imply
  | [ meta fy ; meta v1 ] - [ meta fy ; meta v2 ] | < meta ep end parenthesis end line
line ell n because lemma a4 at meta v2 modus ponens ell m indeed
  0 < meta ep imply meta n2 <= meta v1 imply meta n2 <= meta v2 imply
  | [ meta fy ; meta v1 ] - [ meta fy ; meta v2 ] | < meta ep end line
line ell o because prop lemma mp3 modus ponens ell n modus ponens ell c modus ponens ell big b 
  modus ponens ell i indeed | [ meta fy ; meta v1 ] - [ meta fy ; meta v2 ] | < meta ep end line

because prop lemma join conjuncts modus ponens ell l modus ponens ell o indeed
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < meta ep and0
  | [ meta fy ; meta v1 ] - [ meta fy ; meta v2 ] | < meta ep end line
line ell big a end block

any term meta v1 comma meta v2 comma meta n1 comma meta n2 comma meta ep 
                 comma meta fx comma meta fy end line

because 1rule deduction modus ponens ell big a indeed 
for all meta v1 comma meta v2 indeed parenthesis 0 < meta ep imply 
  meta n1 <= meta v1 imply meta n1 <= meta v2 imply
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < meta ep end parenthesis imply
for all meta v1 comma meta v2 indeed parenthesis 0 < meta ep imply 
  meta n2 <= meta v1 imply meta n2 <= meta v2 imply
  | [ meta fy ; meta v1 ] - [ meta fy ; meta v2 ] | < meta ep end parenthesis imply
0 < meta ep imply max( meta n1 , meta n2 ) <= meta v1 imply max( meta n1 , meta n2 ) <= meta v2 imply
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < meta ep and0
  | [ meta fy ; meta v1 ] - [ meta fy ; meta v2 ] | < meta ep qed
end math ]"

"[  math in theory system Q lemma lemma 2cauchy says
for all terms meta v1 comma meta v2 comma meta n comma meta n1 comma meta n2 comma meta ep 
                      comma meta fx comma meta fy indeed ""; XX lidt grimt med de ekstra variable
for all meta ep indeed exist0 meta n indeed 
  for all meta v1 comma meta v2 indeed parenthesis 0 < meta ep imply 
  meta n <= meta v1 imply meta n <= meta v2 imply
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < meta ep and0
  | [ meta fy ; meta v1 ] - [ meta fy ; meta v2 ] | < meta ep end parenthesis
end lemma end math ]"

"[ math system Q proof of lemma 2cauchy reads

any term meta v1 comma meta v2 comma meta n comma meta n1 comma meta n2 comma meta ep 
                 comma meta fx comma meta fy end line


line ell a because axiom cauchy indeed 
  for all meta ep indeed exist0 meta n1 indeed for all meta v1 comma meta v2 indeed parenthesis
  0 < meta ep imply meta n1 <= meta v1 imply meta n1 <= meta v2 imply 
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < meta ep end parenthesis end line
line ell big a because lemma a4 at meta ep modus ponens ell a indeed
  exist0 meta n1 indeed for all meta v1 comma meta v2 indeed parenthesis
  0 < meta ep imply meta n1 <= meta v1 imply meta n1 <= meta v2 imply 
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < meta ep end parenthesis end line

line ell b because axiom cauchy indeed 
  for all meta ep indeed exist0 meta n2 indeed for all meta v1 comma meta v2 indeed parenthesis
  0 < meta ep imply meta n2 <= meta v1 imply meta n2 <= meta v2 imply 
  | [ meta fy ; meta v1 ] - [ meta fy ; meta v2 ] | < meta ep end parenthesis end line
line ell big b because lemma a4 at meta ep modus ponens ell b indeed
  exist0 meta n2 indeed for all meta v1 comma meta v2 indeed parenthesis
  0 < meta ep imply meta n2 <= meta v1 imply meta n2 <= meta v2 imply 
  | [ meta fy ; meta v1 ] - [ meta fy ; meta v2 ] | < meta ep end parenthesis end line

line ell c because lemma 2cauchy helper indeed
for all meta v1 comma meta v2 indeed parenthesis 0 < meta ep imply 
  meta n1 <= meta v1 imply meta n1 <= meta v2 imply
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < meta ep end parenthesis imply
for all meta v1 comma meta v2 indeed parenthesis 0 < meta ep imply 
  meta n2 <= meta v1 imply meta n2 <= meta v2 imply
  | [ meta fy ; meta v1 ] - [ meta fy ; meta v2 ] | < meta ep end parenthesis imply
0 < meta ep imply max( meta n1 , meta n2 ) <= meta v1 imply max( meta n1 , meta n2 ) <= meta v2 imply 
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < meta ep and0
  | [ meta fy ; meta v1 ] - [ meta fy ; meta v2 ] | < meta ep end line


line ell d because pred lemma exist mp2 modus ponens ell c modus ponens ell big a modus ponens ell big b indeed 0 < meta ep imply max( meta n1 , meta n2 ) <= meta v1 imply max( meta n1 , meta n2 ) <= meta v2 imply
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < meta ep and0
  | [ meta fy ; meta v1 ] - [ meta fy ; meta v2 ] | < meta ep end line

line ell e because 1rule gen modus ponens ell d indeed for all meta v2 indeed parenthesis
  0 < meta ep imply max( meta n1 , meta n2 ) <= meta v1 imply max( meta n1 , meta n2 ) <= meta v2 imply
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < meta ep and0
  | [ meta fy ; meta v1 ] - [ meta fy ; meta v2 ] | < meta ep end parenthesis end line

line ell f because 1rule gen modus ponens ell e indeed for all meta v1 comma meta v2 indeed parenthesis
  0 < meta ep imply max( meta n1 , meta n2 ) <= meta v1 imply max( meta n1 , meta n2 ) <= meta v2 imply
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < meta ep and0
  | [ meta fy ; meta v1 ] - [ meta fy ; meta v2 ] | < meta ep end parenthesis end line

line ell g because pred lemma intro exist at max( meta n1 , meta n2 ) modus ponens ell f indeed 
  exist0 meta n indeed for all meta v1 comma meta v2 indeed parenthesis
  0 < meta ep imply meta n <= meta v1 imply meta n <= meta v2 imply
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < meta ep and0
  | [ meta fy ; meta v1 ] - [ meta fy ; meta v2 ] | < meta ep end parenthesis end line

because 1rule gen modus ponens ell g indeed 
  for all meta ep indeed exist0 meta n indeed for all meta v1 comma meta v2 indeed parenthesis
  0 < meta ep imply meta n <= meta v1 imply meta n <= meta v2 imply
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < meta ep and0
  | [ meta fy ; meta v1 ] - [ meta fy ; meta v2 ] | < meta ep end parenthesis qed
end math ]"



"[  math in theory system Q lemma pred lemma EAE mp says
for all terms meta v1 comma meta v2 comma meta v3 comma meta a comma meta b indeed
meta a imply meta b infer
exist0 meta v1 indeed for all meta v2 indeed exist0 meta v3 indeed meta a infer meta b
end lemma end math ]"

"[ math system Q proof of pred lemma EAE mp reads

block any term meta v2 comma meta v3 comma meta a comma meta b end line
line ell a premise meta a imply meta b end line
line ell b premise for all meta v2 indeed exist0 meta v3 indeed meta a end line
line ell c because lemma a4 at meta v2 modus ponens ell b indeed exist0 meta v3 indeed meta a end line
because pred lemma exist mp modus ponens ell a modus ponens ell c indeed meta b end line
line ell big a end block


any term meta v1 comma meta v2 comma meta v3 comma meta a comma meta b end line

line ell a because 1rule deduction modus ponens ell big a indeed 
  parenthesis meta a imply meta b end parenthesis imply
  for all meta v2 indeed exist0 meta v3 indeed meta a imply meta b end line

line ell b premise meta a imply meta b end line
line ell c premise exist0 meta v1 indeed for all meta v2 indeed exist0 meta v3 indeed meta a end line

line ell d because 1rule mp modus ponens ell a modus ponens ell b indeed
  for all meta v2 indeed exist0 meta v3 indeed meta a imply meta b end line

because pred lemma exist mp modus ponens ell d modus ponens ell c indeed meta b qed
end math ]"







"[  math in theory system Q lemma lemma addEquations(LeqLess) says
for all terms meta x comma meta y comma meta z comma meta u indeed
meta x <= meta y infer meta z < meta u infer
meta x + meta z < meta y + meta u
end lemma end math ]"

"[ math system Q proof of lemma addEquations(LeqLess) reads
any term meta x comma meta y comma meta z comma meta u end line
line ell a premise meta x <= meta y end line
line ell b premise meta z < meta u end line
line ell c because lemma leqAddition modus ponens ell a indeed meta x + meta z <= meta y + meta z end line
line ell d because lemma lessAdditionLeft modus ponens ell b indeed 
  meta y + meta z < meta y + meta u end line

because lemma leqLessTransitivity modus ponens ell c modus ponens ell d indeed 
  meta x + meta z < meta y + meta u qed
end math ]"


"[  math in theory system Q lemma lemma insertTwoMiddleTerms(Numerical) says
for all terms meta x comma meta y comma meta z comma meta u indeed
| meta x + meta y | <= | meta x - meta z | + | meta z - meta u | + | meta u + meta y | 
end lemma end math ]"

"[ math system Q proof of lemma insertTwoMiddleTerms(Numerical) reads
any term meta x comma meta y comma meta z comma meta u end line
line ell a because lemma insertMiddleTerm(Numerical) indeed
  | meta x + meta y | <= | meta x - meta z | + | meta z + meta y | end line

line ell b because lemma insertMiddleTerm(Numerical) indeed
  | meta z + meta y | <= | meta z - meta u | + | meta u + meta y | end line
line ell c because lemma leqAdditionLeft modus ponens ell b indeed
  | meta x - meta z | + | meta z + meta y | <= 
  | meta x - meta z | + parenthesis | meta z - meta u | + | meta u + meta y | end parenthesis end line

line ell d because lemma leqTransitivity modus ponens ell a modus ponens ell c indeed
  | meta x + meta y | <= 
  | meta x - meta z | + parenthesis | meta z - meta u | + | meta u + meta y | end parenthesis end line

line ell e because axiom plusAssociativity indeed
  | meta x - meta z | + | meta z - meta u | + | meta u + meta y | =
  | meta x - meta z | + parenthesis | meta z - meta u | + | meta u + meta y | end parenthesis end line
line ell f because lemma eqSymmetry modus ponens ell e indeed
  | meta x - meta z | + parenthesis | meta z - meta u | + | meta u + meta y | end parenthesis =
  | meta x - meta z | + | meta z - meta u | + | meta u + meta y | end line
  
because lemma subLeqRight modus ponens ell f modus ponens ell d indeed 
  | meta x + meta y | <= | meta x - meta z | + | meta z - meta u | + | meta u + meta y | qed
end math ]"

"[  math in theory system Q lemma lemma fromPositiveNumerical says
for all terms meta x indeed
0 < | meta x | infer meta x != 0
end lemma end math ]"

"[ math system Q proof of lemma fromPositiveNumerical reads

block any term meta x end line
line ell b premise 0 <= meta x end line
line ell a premise 0 < | meta x | end line
line ell c because lemma nonnegativeNumerical modus ponens ell b indeed | meta x | = meta x end line
line ell d because lemma subLessRight modus ponens ell c modus ponens ell a indeed 0 < meta x end line
line ell e because lemma lessNeq modus ponens ell d indeed 0 != meta x end line
because lemma neqSymmetry modus ponens ell e indeed meta x != 0 end line
 line ell big a end block

block any term meta x end line
line ell b premise meta x <= 0 end line
line ell a premise 0 < | meta x | end line
line ell c because lemma nonpositiveNumerical modus ponens ell b indeed | meta x | = - meta x end line
line ell d because lemma subLessRight modus ponens ell c modus ponens ell a indeed 0 < - meta x end line
line ell e because lemma positiveNegated modus ponens ell d indeed - - meta x < 0 end line

line ell f because lemma doubleMinus indeed - - meta x = meta x end line
line ell g because lemma subLessLeft modus ponens ell f modus ponens ell e indeed meta x < 0 end line
because lemma lessNeq modus ponens ell g indeed meta x != 0 end line
line ell big b end block

any term meta x end line

line ell a because 1rule deduction modus ponens ell big a indeed 
  0 <= meta x imply 0 < | meta x | imply meta x != 0 end line

line ell b because 1rule deduction modus ponens ell big b indeed 
  meta x <= 0 imply 0 < | meta x | imply meta x != 0 end line

line ell c premise 0 < | meta x | end line

line ell d because lemma from leqGeq modus ponens ell a modus ponens ell b indeed 0 < | meta x | imply meta x != 0 end line  

because 1rule mp modus ponens ell d modus ponens ell c indeed meta x != 0 qed
end math ]"



"[  math in theory system Q lemma lemma negativeToRight(Neq)(1 term) says
for all terms meta x comma meta y indeed
meta x - meta y != 0 infer meta x != meta y
end lemma end math ]"

"[ math system Q proof of lemma negativeToRight(Neq)(1 term) reads

block any term meta x comma meta y end line
line ell a premise meta x = meta y end line
because lemma positiveToLeft(Eq)(1 term) modus ponens ell a indeed meta x - meta y = 0 end line
line ell big a end block

any term meta x comma meta y end line

line ell a because 1rule deduction modus ponens ell big a indeed 
  meta x = meta y imply meta x - meta y = 0 end line

line ell b premise meta x - meta y != 0 end line

because prop lemma mt modus ponens ell a modus ponens ell b indeed meta x != meta y qed
end math ]"


"[  math in theory system Q lemma lemma nonzeroProduct(2) says
for all terms meta x comma meta y indeed
meta x * meta y != 0 infer meta y != 0
end lemma end math ]"

"[ math system Q proof of lemma nonzeroProduct(2) reads
block any term meta x comma meta y end line
line ell a premise meta y = 0 end line
line ell b because lemma eqMultiplicationLeft modus ponens ell a indeed 
  meta x * meta y = meta x * 0 end line
line ell c because lemma x*0=0 indeed meta x * 0 = 0 end line
because lemma eqTransitivity modus ponens ell b modus ponens ell c indeed meta x * meta y = 0 end line
line ell big a end block

any term meta x comma meta y end line

line ell a because 1rule deduction modus ponens ell big a indeed 
  meta y = 0 imply meta x * meta y = 0 end line

line ell b premise meta x * meta y != 0 end line

because prop lemma mt modus ponens ell a modus ponens ell b indeed meta y != 0 qed
end math ]"


"[  math in theory system Q lemma lemma nonreciprocalToRight(Eq)(1 term) says
for all terms meta x comma meta y indeed
meta x * meta y = 1 infer
meta x = 1/ meta y 
end lemma end math ]"


"[ math system Q proof of lemma nonreciprocalToRight(Eq)(1 term) reads
any term meta x comma meta y end line
line ell a premise meta x * meta y = 1 end line
line ell b because lemma eqMultiplication modus ponens ell a indeed
  meta x * meta y * 1/ meta y = 1 * 1/ meta y end line

line ell c because lemma 0<1 indeed 0 < 1 end line
line ell d because lemma positiveNonzero modus ponens ell c indeed 1 != 0 end line
line ell e because lemma eqSymmetry modus ponens ell a indeed 1 = meta x * meta y end line
line ell f because lemma subNeqLeft modus ponens ell e modus ponens ell d indeed 
  meta x * meta y != 0 end line
line ell g because lemma nonzeroProduct(2) modus ponens ell f indeed meta y != 0 end line

line ell h because lemma x=x*y*(1/y) modus ponens ell g indeed 
  meta x = meta x * meta y * 1/ meta y end line

line ell i because lemma times1Left indeed 1 * 1/ meta y = 1/ meta y end line

because lemma eqTransitivity4 modus ponens ell h modus ponens ell b modus ponens ell i indeed 
  meta x = 1/ meta y qed
end math ]"


"[  math in theory system Q lemma lemma nonreciprocalToRight(Eq) says
for all terms meta x comma meta y comma meta z indeed
meta y != 0 infer
meta x * meta y = meta z infer
meta x = meta z * 1/ meta y 
end lemma end math ]"


"[ math system Q proof of lemma nonreciprocalToRight(Eq) reads
any term meta x comma meta y comma meta z end line
line ell g premise meta y != 0 end line
line ell a premise meta x * meta y = meta z end line
line ell b because lemma eqMultiplication modus ponens ell a indeed
  meta x * meta y * 1/ meta y = meta z * 1/ meta y end line


line ell h because lemma x=x*y*(1/y) modus ponens ell g indeed 
  meta x = meta x * meta y * 1/ meta y end line

because lemma eqTransitivity modus ponens ell h modus ponens ell b indeed 
  meta x = meta z * 1/ meta y qed
end math ]"


"[ math in theory system Q lemma lemma nonreciprocalToLeft(Eq)(1 term) says
for all terms meta x comma meta y indeed
1 = meta x * meta y infer 1/ meta y = meta x
end lemma end math ]"

"[ math system Q proof of lemma nonreciprocalToLeft(Eq)(1 term) reads
any term meta x comma meta y end line
line ell b premise 1 = meta x * meta y end line
line ell c because lemma eqSymmetry modus ponens ell b indeed meta x * meta y = 1 end line
line ell d because lemma nonreciprocalToRight(Eq)(1 term) modus ponens ell c indeed
  meta x = 1/ meta y end line
because lemma eqSymmetry modus ponens ell d indeed 1/ meta y = meta x qed
end math ]"

"[ math in theory system Q lemma lemma sameReciprocal says
for all terms meta x comma meta y indeed
meta x != 0 infer meta x = meta y infer 1/ meta x = 1/ meta y 
end lemma end math ]"

"[ math system Q proof of lemma sameReciprocal reads
any term meta x comma meta y end line
line ell a premise meta x != 0 end line
line ell b premise meta x = meta y end line
line ell c because lemma times1Left indeed 1 * meta x = meta x end line
line ell d because lemma eqTransitivity modus ponens ell c modus ponens ell b indeed 
  1 * meta x = meta y end line
line ell e because lemma nonreciprocalToRight(Eq) modus ponens ell a modus ponens ell d indeed
  1 = meta y * 1/ meta x end line
line ell f because axiom timesCommutativity indeed meta y * 1/ meta x = 1/ meta x * meta y end line
line ell g because lemma eqTransitivity modus ponens ell e modus ponens ell f indeed


  1 = 1/ meta x * meta y end line
line ell h because lemma nonreciprocalToLeft(Eq)(1 term) modus ponens ell g indeed
  1/ meta y = 1/ meta x end line
because lemma eqSymmetry modus ponens ell h indeed 1/ meta x = 1/ meta y qed
end math ]"



"[ math in theory system Q lemma lemma orderedPairEquality says
for all terms meta sx comma meta sx1 comma meta sy comma meta sy1 comma meta sz comma meta sz1 comma
meta su comma meta su1 indeed
(o meta sx , meta sx1 ) = (o meta sy , meta sy1 ) infer
(o meta sz , meta sz1 ) = (o meta su , meta su1 ) infer
meta sx = meta sz infer meta sy = meta su
end lemma end math ]"

"[ math system Q proof of lemma orderedPairEquality reads
any term meta sx comma meta sx1 comma meta sy comma meta sy1 comma meta sz comma meta sz1 comma
meta su comma meta su1 end line
line ell a premise (o meta sx , meta sx1 ) = (o meta sy , meta sy1 ) end line
line ell b premise (o meta sz , meta sz1 ) = (o meta su , meta su1 ) end line
line ell c premise meta sx = meta sz end line
line ell d because lemma fromOrderedPair(1) modus ponens ell a indeed meta sx = meta sy end line
line ell e because lemma eqSymmetry modus ponens ell d indeed meta sy = meta sx end line
line ell f because lemma fromOrderedPair(1) modus ponens ell b indeed meta sz = meta su end line
because lemma eqTransitivity4 modus ponens ell e modus ponens ell c modus ponens ell f indeed 
  meta sy = meta su qed
end math ]"




"[ math in theory system Q lemma lemma reciprocalIsFunction says
  for all terms meta m comma meta m comma meta fx indeed ""; XX ekstra variable
  for all object f1 comma object f2 comma object f3 comma object f4 indeed
  ( (o object f1 , object f2 ) in0 1fny/ meta fx imply (o object f3 , object f4 ) in0 1fny/ meta fx imply
  object f1 = object f3 imply object f2 = object f4 )
end lemma end math ]"

"[ math system Q proof of lemma reciprocalIsFunction reads 
block any term meta m comma meta m comma meta fx end line
line ell a premise (o object f1 , object f2 ) in0 1fny/ meta fx end line
line ell b premise (o object f3 , object f4 ) in0 1fny/ meta fx end line
line ell c premise object f1 = object f3 end line

line ell d because 1rule repetition modus ponens ell a indeed ""; XXREP 1fny/
  (o object f1 , object f2 ) in0 the set of ph in cartProd( N , Q ) such that exist0 meta m indeed
    ( [ meta fx ; meta m ] != 0 and0  ph6 = (o meta m , 1/ [ meta fx ; meta m ] ) ) or0 
    ( [ meta fx ; meta m ] = 0 and0 ph6 = (o meta m , 0 ) ) end set end line
line ell e because lemma separation2formula(2) modus ponens ell d indeed  exist0 meta m indeed
( [ meta fx ; meta m ] != 0 and0 (o object f1 , object f2 ) = (o meta m , 1/ [ meta fx ; meta m ] ) ) or0 
  ( [ meta fx ; meta m ]  = 0 and0 (o object f1 , object f2 ) = (o meta m , 0 ) ) end line
  
line ell f because 1rule repetition modus ponens ell b indeed ""; XXREP 1fny/
  (o object f3 , object f4 ) in0 the set of ph in cartProd( N , Q ) such that exist0 meta m indeed
    ( [ meta fx ; meta m ] != 0 and0  ph6 = (o meta m , 1/ [ meta fx ; meta m ] ) ) or0 
    ( [ meta fx ; meta m ] = 0 and0 ph6 = (o meta m , 0 ) ) end set end line
line ell g because lemma separation2formula(2) modus ponens ell f indeed  exist0 meta m indeed
( [ meta fx ; meta m ] != 0 and0 (o object f1 , object f2 ) = (o meta m , 1/ [ meta fx ; meta m ] ) ) or0 
( [ meta fx ; meta m ]  = 0 and0 (o object f1 , object f2 ) = (o meta m , 0 ) ) end line

block any term meta m comma meta m comma meta fx end line
line ell c premise object f1 = object f3 end line
line ell a premise [ meta fx ; meta m ] != 0 and0 
                   (o object f1 , object f2 ) = (o meta m , 1/ [ meta fx ; meta m ] ) end line
line ell b premise [ meta fx ; meta m ] != 0 and0 
                   (o object f3 , object f4 ) = (o meta m , 1/ [ meta fx ; meta m ] ) end line
line ell d because prop lemma first conjunct modus ponens ell a indeed [ meta fx ; meta m ] != 0 end line
line ell e because prop lemma second conjunct modus ponens ell a indeed 
  (o object f1 , object f2 ) = (o meta m , 1/ [ meta fx ; meta m ] ) end line
line ell f because prop lemma second conjunct modus ponens ell b indeed
  (o object f3 , object f4 ) = (o meta m , 1/ [ meta fx ; meta m ] ) end line

line ell g because lemma fromOrderedPair(1) modus ponens ell e indeed object f1 = meta m end line
line ell h because lemma eqSymmetry modus ponens ell g indeed meta m = object f1 end line
line ell i because lemma fromOrderedPair(1) modus ponens ell f indeed object f3 = meta m end line
line ell j because lemma eqTransitivity4 modus ponens ell h modus ponens ell c modus ponens ell i indeed
  meta m = meta m end line

line ell k because lemma sameSeries modus ponens ell j indeed 
  [ meta fx ; meta m ] = [ meta fx ; meta m ] end line
line ell l because lemma sameReciprocal modus ponens ell d modus ponens ell k indeed
  1/ [ meta fx ; meta m ] = 1/ [ meta fx ; meta m ] end line

line ell m because lemma fromOrderedPair(2) modus ponens ell e indeed 
  object f2 = 1/ [ meta fx ; meta m ] end line
line ell n because lemma fromOrderedPair(2) modus ponens ell f indeed 
  object f4 = 1/ [ meta fx ; meta m ] end line
line ell o because lemma eqSymmetry modus ponens ell n indeed
  1/ [ meta fx ; meta m ] = object f4 end line  
because lemma eqTransitivity4 modus ponens ell m modus ponens ell l modus ponens ell o indeed
  object f2 = object f4 end line
line ell big x end block

block any term meta m comma meta m comma meta fx end line
line ell c premise object f1 = object f3 end line
line ell a premise [ meta fx ; meta m ] != 0 and0 
                   (o object f1 , object f2 ) = (o meta m , 1/ [ meta fx ; meta m ] ) end line
line ell b premise [ meta fx ; meta m ] = 0 and0 
                   (o object f3 , object f4 ) = (o meta m , 0 ) end line
line ell d because prop lemma first conjunct modus ponens ell a indeed [ meta fx ; meta m ] != 0 end line
line ell e because prop lemma second conjunct modus ponens ell a indeed
  (o object f1 , object f2 ) = (o meta m , 1/ [ meta fx ; meta m ] ) end line
line ell f because prop lemma first conjunct modus ponens ell b indeed [ meta fx ; meta m ] = 0 end line
line ell g because prop lemma second conjunct modus ponens ell b indeed
  (o object f3 , object f4 ) = (o meta m , 0 ) end line
line ell h because lemma orderedPairEquality modus ponens ell e modus ponens ell g modus ponens ell c indeed
  meta m = meta m end line
line ell i because lemma sameSeries modus ponens ell h indeed 
  [ meta fx ; meta m ] = [ meta fx ; meta m ] end line 
line ell j because lemma eqTransitivity modus ponens ell i modus ponens ell f indeed 
  [ meta fx ; meta m ] = 0 end line
because prop lemma from contradiction modus ponens ell j modus ponens ell d indeed
  object f2 = object f4 end line  
line ell big y end block


block any term meta m comma meta m comma meta fx end line
line ell c premise object f1 = object f3 end line
line ell a premise [ meta fx ; meta m ] = 0 and0 
                   (o object f1 , object f2 ) = (o meta m , 0 ) end line
line ell b premise [ meta fx ; meta m ] != 0 and0 
                   (o object f3 , object f4 ) = (o meta m , 1/ [ meta fx ; meta m ] ) end line
line ell d because prop lemma first conjunct modus ponens ell a indeed [ meta fx ; meta m ] = 0 end line
line ell e because prop lemma second conjunct modus ponens ell a indeed
  (o object f1 , object f2 ) = (o meta m , 0 ) end line
line ell f because prop lemma first conjunct modus ponens ell b indeed [ meta fx ; meta m ] != 0 end line
line ell g because prop lemma second conjunct modus ponens ell b indeed

  (o object f3 , object f4 ) = (o meta m , 1/ [ meta fx ; meta m ] ) end line
line ell h because lemma orderedPairEquality modus ponens ell e modus ponens ell g modus ponens ell c indeed
  meta m = meta m end line
line ell i because lemma sameSeries modus ponens ell h indeed 
  [ meta fx ; meta m ] = [ meta fx ; meta m ] end line 
line ell j because lemma eqSymmetry modus ponens ell i indeed  
  [ meta fx ; meta m ] = [ meta fx ; meta m ] end line 
line ell k because lemma eqTransitivity modus ponens ell j modus ponens ell d indeed 
  [ meta fx ; meta m ] = 0 end line
because prop lemma from contradiction modus ponens ell k modus ponens ell f indeed
  object f2 = object f4 end line  
line ell big z end block

block any term meta m comma meta m comma meta fx end line
line ell a premise [ meta fx ; meta m ] = 0 and0 
                   (o object f1 , object f2 ) = (o meta m , 0 ) end line
line ell b premise [ meta fx ; meta m ] = 0 and0 
                   (o object f3 , object f4 ) = (o meta m , 0 ) end line
line ell c because prop lemma second conjunct modus ponens ell a indeed 
  (o object f1 , object f2 ) = (o meta m , 0 ) end line
line ell k because prop lemma second conjunct modus ponens ell b indeed
  (o object f3 , object f4 ) = (o meta m , 0 ) end line
line ell d because lemma fromOrderedPair(2) modus ponens ell c indeed object f2 = 0 end line
line ell e because lemma fromOrderedPair(2) modus ponens ell k indeed object f4 = 0 end line
line ell f because lemma eqSymmetry modus ponens ell e indeed 0 = object f4 end line
because lemma eqTransitivity modus ponens ell d modus ponens ell f indeed object f2 = object f4 end line
line ell big u end block

block any term meta m comma meta m comma meta fx end line
line ell a premise object f1 = object f3 end line
line ell b premise 
( [ meta fx ; meta m ] != 0 and0 (o object f1 , object f2 ) = (o meta m , 1/ [ meta fx ; meta m ] ) ) or0 
  ( [ meta fx ; meta m ]  = 0 and0 (o object f1 , object f2 ) = (o meta m , 0 ) ) end line
line ell c premise 
( [ meta fx ; meta m ] != 0 and0 (o object f3 , object f4 ) = (o meta m , 1/ [ meta fx ; meta m ] ) ) or0 
( [ meta fx ; meta m ]  = 0 and0 (o object f3 , object f4 ) = (o meta m , 0 ) ) end line

line ell d because 1rule deduction modus ponens ell big x indeed
object f1 = object f3 imply
[ meta fx ; meta m ] != 0 and0 
  (o object f1 , object f2 ) = (o meta m , 1/ [ meta fx ; meta m ] ) imply
[ meta fx ; meta m ] != 0 and0 
  (o object f3 , object f4 ) = (o meta m , 1/ [ meta fx ; meta m ] ) imply object f2 = object f4 end line
line ell e because 1rule mp modus ponens ell d modus ponens ell a indeed
[ meta fx ; meta m ] != 0 and0 
  (o object f1 , object f2 ) = (o meta m , 1/ [ meta fx ; meta m ] ) imply
[ meta fx ; meta m ] != 0 and0 
  (o object f3 , object f4 ) = (o meta m , 1/ [ meta fx ; meta m ] ) imply object f2 = object f4 end line

line ell f because 1rule deduction modus ponens ell big y indeed
object f1 = object f3 imply
[ meta fx ; meta m ] != 0 and0 
  (o object f1 , object f2 ) = (o meta m , 1/ [ meta fx ; meta m ] ) imply
[ meta fx ; meta m ] = 0 and0 
  (o object f3 , object f4 ) = (o meta m , 0 ) imply object f2 = object f4 end line
line ell g because 1rule mp modus ponens ell f modus ponens ell a indeed
[ meta fx ; meta m ] != 0 and0 
  (o object f1 , object f2 ) = (o meta m , 1/ [ meta fx ; meta m ] ) imply
[ meta fx ; meta m ] = 0 and0 
  (o object f3 , object f4 ) = (o meta m , 0 ) imply object f2 = object f4 end line

line ell h because 1rule deduction modus ponens ell big z indeed
object f1 = object f3 imply
[ meta fx ; meta m ] = 0 and0 
  (o object f1 , object f2 ) = (o meta m , 0 ) imply
[ meta fx ; meta m ] != 0 and0 
  (o object f3 , object f4 ) = (o meta m , 1/ [ meta fx ; meta m ] ) imply object f2 = object f4 end line
line ell i because 1rule mp modus ponens ell h modus ponens ell a indeed
[ meta fx ; meta m ] = 0 and0 
  (o object f1 , object f2 ) = (o meta m , 0 ) imply
[ meta fx ; meta m ] != 0 and0 
  (o object f3 , object f4 ) = (o meta m , 1/ [ meta fx ; meta m ] ) imply object f2 = object f4 end line

line ell j because 1rule deduction modus ponens ell big u indeed
[ meta fx ; meta m ] = 0 and0 
  (o object f1 , object f2 ) = (o meta m , 0 ) imply
[ meta fx ; meta m ] = 0 and0 
  (o object f3 , object f4 ) = (o meta m , 0 ) imply object f2 = object f4 end line

 because prop lemma from two times two disjuncts modus ponens ell b modus ponens ell c
  modus ponens ell e modus ponens ell g modus ponens ell i modus ponens ell j indeed 
  object f2 = object f4 end line

line ell big v end block

line ell h because 1rule deduction modus ponens ell big v indeed
object f1 = object f3 imply
( [ meta fx ; meta m ] != 0 and0 (o object f1 , object f2 ) = (o meta m , 1/ [ meta fx ; meta m ] ) ) or0 
  ( [ meta fx ; meta m ]  = 0 and0 (o object f1 , object f2 ) = (o meta m , 0 ) ) imply
( [ meta fx ; meta m ] != 0 and0 (o object f1 , object f2 ) = (o meta m , 1/ [ meta fx ; meta m ] ) ) or0 
  ( [ meta fx ; meta m ]  = 0 and0 (o object f1 , object f2 ) = (o meta m , 0 ) ) imply 
object f2 = object f4 end line

line ell i because 1rule mp modus ponens ell h modus ponens ell c indeed 
( [ meta fx ; meta m ] != 0 and0 (o object f1 , object f2 ) = (o meta m , 1/ [ meta fx ; meta m ] ) ) or0 
  ( [ meta fx ; meta m ]  = 0 and0 (o object f1 , object f2 ) = (o meta m , 0 ) ) imply
( [ meta fx ; meta m ] != 0 and0 (o object f1 , object f2 ) = (o meta m , 1/ [ meta fx ; meta m ] ) ) or0 
  ( [ meta fx ; meta m ]  = 0 and0 (o object f1 , object f2 ) = (o meta m , 0 ) ) imply 
object f2 = object f4 end line


because pred lemma exist mp2 modus ponens ell i modus ponens ell e modus ponens ell g indeed
object f2 = object f4 end line
line ell big a end block

any term meta m comma meta m comma meta fx end line

because 1rule deduction modus ponens ell big a indeed  
  for all object f1 comma object f2 comma object f3 comma object f4 indeed
  ( (o object f1 , object f2 ) in0 1fny/ meta fx imply (o object f3 , object f4 ) in0 1fny/ meta fx imply
  object f1 = object f3 imply object f2 = object f4 ) qed
end math ]"

"[ math in theory system Q lemma lemma reciprocalIsTotal says 
for all terms meta fx indeed 
for all object s1 indeed 
  ( object s1 in0 N imply exist0 object s2 indeed (o object s1 , object s2 ) in0 1fny/ meta fx )
end lemma end math ]"

"[ math system Q proof of lemma reciprocalIsTotal reads
block any term meta fx end line
line ell b premise [ meta fx ; object s1 ] = 0 end line
line ell a premise object s1 in0 N end line
line ell c because axiom rationalType indeed 0 in0 Q end line
line ell d because lemma toCartProd modus ponens ell a modus ponens ell c indeed
  (o object s1 , 0 ) in0 cartProd( N , Q ) end line
line ell e because lemma eqReflexivity indeed (o object s1 , 0 ) = (o object s1 , 0 ) end line
line ell f because prop lemma join conjuncts modus ponens ell b modus ponens ell e indeed
  [ meta fx ; object s1 ] = 0 and0 (o object s1 , 0 ) = (o object s1 , 0 ) end line
line ell g because prop lemma weaken or first modus ponens ell f indeed
  ( [ meta fx ; object s1 ] != 0 and0 (o object s1 , 0 ) = (o object s1 , 1/ [ meta fx ; object s1 ] ) ) or0
  ( [ meta fx ; object s1 ] = 0  and0 (o object s1 , 0 ) = (o object s1 , 0 ) ) end line


line ell h because pred lemma intro exist at object s1 modus ponens ell g indeed exist0 meta m indeed 
  ( [ meta fx ; meta m ] != 0  and0 (o object s1 , 0 ) = (o meta m , 1/ [ meta fx ; meta m ] ) ) or0 
  ( [ meta fx ; meta m ]  = 0  and0 (o object s1 , 0 ) = (o meta m , 0 ) ) end line

line ell i because lemma formula2separation modus ponens ell d modus ponens ell h indeed
  (o object s1 , 0 ) in0 1fny/ meta fx end line

because pred lemma intro exist at 0 modus ponens ell i indeed
  exist0 object s2 indeed (o object s1 , object s2 ) in0 1fny/ meta fx end line
line ell big a end block

block any term meta fx end line
line ell b premise [ meta fx ; object s1 ] != 0 end line
line ell a premise object s1 in0 N end line
line ell big e because axiom seriesType indeed isSeries( meta fx , Q ) end line
line ell big f because lemma valueType modus ponens ell a modus ponens ell big e indeed
  [ meta fx ; object s1 ] in0 Q end line

line ell c because lemma QisClosed(reciprocal) modus ponens ell b modus ponens ell big f indeed 
   1/ [ meta fx ; object s1 ] in0 Q end line
line ell d because lemma toCartProd modus ponens ell a modus ponens ell c indeed
  (o object s1 , 1/ [ meta fx ; object s1 ] ) in0 cartProd( N , Q ) end line
line ell e because lemma eqReflexivity indeed 
  (o object s1 , 1/ [ meta fx ; object s1 ] ) = (o object s1 , 1/ [ meta fx ; object s1 ] ) end line
line ell f because prop lemma join conjuncts modus ponens ell b modus ponens ell e indeed
  [ meta fx ; object s1 ] != 0 and0 
  (o object s1 , 1/ [ meta fx ; object s1 ] ) = (o object s1 , 1/ [ meta fx ; object s1 ] ) end line
line ell g because prop lemma weaken or second modus ponens ell f indeed
  ( [ meta fx ; object s1 ] != 0 and0 
    (o object s1 , 1/ [ meta fx ; object s1 ] ) = (o object s1 , 1/ [ meta fx ; object s1 ] ) ) or0
  ( [ meta fx ; object s1 ] = 0  and0 
    (o object s1 , 1/ [ meta fx ; object s1 ] ) = (o object s1 , 0 ) ) end line
line ell h because pred lemma intro exist at object s1 modus ponens ell g indeed exist0 meta m indeed
  ( [ meta fx ; meta m ] != 0 and0 
    (o object s1 , 1/ [ meta fx ; object s1 ] ) = (o meta m , 1/ [ meta fx ; meta m ] ) ) or0
  ( [ meta fx ; meta m ] = 0  and0 
    (o object s1 , 1/ [ meta fx ; object s1 ] ) = (o meta m , 0 ) ) end line
line ell i because lemma formula2separation modus ponens ell d modus ponens ell h indeed
  (o object s1 , 1/ [ meta fx ; object s1 ] ) in0 1fny/ meta fx end line
because pred lemma intro exist at 1/ [ meta fx ; object s1 ] modus ponens ell i indeed
  exist0 object s2 indeed (o object s1 , object s2 ) in0 1fny/ meta fx end line
line ell big b end block

any term meta fx end line

line ell big c because 1rule deduction modus ponens ell big a indeed
  [ meta fx ; object s1 ] = 0 imply for all object s1 indeed ( object s1 in0 N imply 
  exist0 object s2 indeed (o object s1 , object s2 ) in0 1fny/ meta fx ) end line
line ell big d because 1rule deduction modus ponens ell big b indeed
  [ meta fx ; object s1 ] != 0 imply for all object s1 indeed ( object s1 in0 N imply 
  exist0 object s2 indeed (o object s1 , object s2 ) in0 1fny/ meta fx ) end line
because prop lemma from negations modus ponens ell big c modus ponens ell big d indeed
  for all object s1 indeed ( object s1 in0 N imply 
  exist0 object s2 indeed (o object s1 , object s2 ) in0 1fny/ meta fx ) qed
end math ]"






"[ math in theory system Q lemma lemma reciprocalIsRationalSeries says
for all terms meta fx indeed
isSeries( 1fny/ meta fx , Q )
end lemma end math ]"

"[ math system Q proof of lemma reciprocalIsRationalSeries reads
any term meta fx end line
line ell a because lemma CPseparationIsRelation indeed
  isRelation( 1fny/ meta fx , N , Q ) end line
line ell b because lemma reciprocalIsFunction indeed   
  for all object f1 comma object f2 comma object f3 comma object f4 indeed
  ( (o object f1 , object f2 ) in0 1fny/ meta fx imply (o object f3 , object f4 ) in0 1fny/ meta fx imply
  object f1 = object f3 imply object f2 = object f4 ) end line
line ell c because lemma reciprocalIsTotal indeed for all object s1 indeed 
  ( object s1 in0 N imply exist0 object s2 indeed (o object s1 , object s2 ) in0 1fny/ meta fx ) end line

because lemma toSeries modus ponens ell a modus ponens ell b modus ponens ell c indeed 
isSeries( 1fny/ meta fx , Q ) qed
end math ]"

"[  math in theory system Q lemma lemma reciprocalFny nonzero says
for all terms meta m comma meta m1 comma meta fx indeed ""; XX ekstra variable
[ meta fx ; meta m ] != 0 infer [ 1fny/ meta fx ; meta m ] = 1/ [ meta fx ; meta m ]
end lemma end math ]"


"[ math system Q proof of lemma reciprocalFny nonzero reads
any term meta m comma meta m1 comma meta fx end line
line ell a premise [ meta fx ; meta m ] != 0 end line
line ell x because axiom natType indeed meta m in0 N end line
line ell b because lemma reciprocalIsRationalSeries indeed isSeries( 1fny/ meta fx , Q ) end line
line ell c because lemma memberOfSeries modus ponens ell x modus ponens ell b indeed
  (o meta m , [ 1fny/ meta fx ; meta m ] ) in0 1fny/ meta fx end line
line ell d because 1rule repetition modus ponens ell c indeed ""; XXREP 1fny/
  (o meta m , [ 1fny/ meta fx ; meta m ] ) in0 the set of ph in cartProd( N , Q ) such that
  exist0 meta m indeed ( [ meta fx ; meta m ] != 0 and0 ph6 = (o meta m , 1/ [ meta fx ; meta m ] ) ) or0 
  ( [ meta fx ; meta m ] = 0 and0 ph6 = (o meta m , 0 )  ) end set end line
line ell e because lemma separation2formula(2) modus ponens ell d indeed exist0 meta m1 indeed 
  ( [ meta fx ; meta m1 ] != 0 and0 
    (o meta m , [ 1fny/ meta fx ; meta m ] ) = (o meta m1 , 1/ [ meta fx ; meta m1 ] ) ) or0 
  ( [ meta fx ; meta m1 ] = 0 and0 (o meta m , [ 1fny/ meta fx ; meta m ] ) = (o meta m1 , 0 ) ) end line

block any term meta m comma meta m1 comma meta fx end line
line ell a premise [ meta fx ; meta m1 ] != 0 end line
line ell b premise ( [ meta fx ; meta m1 ] != 0 and0 
    (o meta m , [ 1fny/ meta fx ; meta m ] ) = (o meta m1 , 1/ [ meta fx ; meta m1 ] ) ) or0 
  ( [ meta fx ; meta m1 ] = 0 and0 (o meta m , [ 1fny/ meta fx ; meta m ] ) = (o meta m1 , 0 ) ) end line
line ell c because prop lemma to negated and(1) modus ponens ell a indeed
  not0 ( [ meta fx ; meta m1 ] = 0 and0 (o meta m , [ 1fny/ meta fx ; meta m ] ) = (o meta m1 , 0 ) ) end line
line ell d because prop lemma negate second disjunct modus ponens ell b modus ponens ell c indeed
  [ meta fx ; meta m1 ] != 0 and0 
  (o meta m , [ 1fny/ meta fx ; meta m ] ) = (o meta m1 , 1/ [ meta fx ; meta m1 ] ) end line
line ell e because prop lemma second conjunct modus ponens ell d indeed
  (o meta m , [ 1fny/ meta fx ; meta m ] ) = (o meta m1 , 1/ [ meta fx ; meta m1 ] ) end line

line ell j because lemma fromOrderedPair(2) modus ponens ell e indeed 
  [ 1fny/ meta fx ; meta m ] = 1/ [ meta fx ; meta m1 ] end line

line ell f because lemma fromOrderedPair(1) modus ponens ell e indeed meta m = meta m1 end line
line ell g because lemma sameSeries modus ponens ell f indeed 
  [ meta fx ; meta m ] = [ meta fx ; meta m1 ] end line
line ell h because lemma eqSymmetry modus ponens ell g indeed
  [ meta fx ; meta m1 ] = [ meta fx ; meta m ] end line
line ell i because lemma sameReciprocal modus ponens ell a modus ponens ell h indeed
  1/ [ meta fx ; meta m1 ] = 1/ [ meta fx ; meta m ] end line
because lemma eqTransitivity modus ponens ell j modus ponens ell i indeed
   [ 1fny/ meta fx ; meta m ] = 1/ [ meta fx ; meta m ] end line 
line ell big a end block

line ell f because 1rule deduction modus ponens ell big a indeed
[ meta fx ; meta m ] != 0 imply
( [ meta fx ; meta m1 ] != 0 and0 
  (o meta m , [ 1fny/ meta fx ; meta m ] ) = (o meta m1 , 1/ [ meta fx ; meta m1 ] ) ) or0 
  ( [ meta fx ; meta m1 ] = 0 and0 (o meta m , [ 1fny/ meta fx ; meta m ] ) = (o meta m1 , 0 ) ) imply
   [ 1fny/ meta fx ; meta m ] = 1/ [ meta fx ; meta m ] end line 

line ell g because 1rule mp modus ponens ell f modus ponens ell a indeed 
( [ meta fx ; meta m1 ] != 0 and0 
  (o meta m , [ 1fny/ meta fx ; meta m ] ) = (o meta m1 , 1/ [ meta fx ; meta m1 ] ) ) or0 
  ( [ meta fx ; meta m1 ] = 0 and0 (o meta m , [ 1fny/ meta fx ; meta m ] ) = (o meta m1 , 0 ) ) imply
   [ 1fny/ meta fx ; meta m ] = 1/ [ meta fx ; meta m ] end line 

because pred lemma exist mp modus ponens ell g modus ponens ell e indeed 
   [ 1fny/ meta fx ; meta m ] = 1/ [ meta fx ; meta m ] qed
end math ]"




"[  math in theory system Q lemma lemma reciprocalF nonzero says
for all terms meta m comma meta fx indeed
[ meta fx ; meta m ] != 0 infer [ 1f/ meta fx ; meta m ] = 1/ [ meta fx ; meta m ]
end lemma end math ]"

"[ math system Q proof of lemma reciprocalF nonzero reads
any term meta m comma meta fx end line
line ell a premise [ meta fx ; meta m ] != 0 end line
line ell b because 1rule ifThenElse false modus ponens ell a indeed
  if( [ meta fx ; meta m ] = 0 , 0 , 1/ [ meta fx ; meta m ] ) = 1/ [ meta fx ; meta m ] end line
line ell c because axiom reciprocalF indeed
  [ 1f/ meta fx ; meta m ] = if( [ meta fx ; meta m ] = 0 , 0 , 1/ [ meta fx ; meta m ] ) end line

because lemma eqTransitivity modus ponens ell c modus ponens ell b indeed 
  [ 1f/ meta fx ; meta m ] = 1/ [ meta fx ; meta m ] qed
end math ]"



"[  math in theory system Q lemma lemma eventually=f to sameF helper says
for all terms meta m comma meta n comma meta ep comma meta fx comma meta fy indeed
for all meta m indeed 
parenthesis meta n <= meta m imply [ meta fx ; meta m ] = [ meta fy ; meta m ] end parenthesis imply
0 < meta ep imply meta n <= meta m imply
| [ meta fx ; meta m ] - [ meta fy ; meta m ] | < meta ep  
end lemma end math ]"

"[ math system Q proof of lemma eventually=f to sameF helper reads
block any term meta m comma meta n comma meta ep comma meta fx comma meta fy end line
line ell big b premise for all meta m indeed 
  parenthesis meta n <= meta m imply [ meta fx ; meta m ] = [ meta fy ; meta m ] end parenthesis end line
line ell a because lemma a4 at meta m modus ponens ell big b indeed
   meta n <= meta m imply [ meta fx ; meta m ] = [ meta fy ; meta m ] end line
line ell b premise 0 < meta ep end line
line ell c premise meta n <= meta m end line
line ell d because 1rule mp modus ponens ell a modus ponens ell c indeed 
  [ meta fx ; meta m ] = [ meta fy ; meta m ] end line
line ell e because lemma positiveToLeft(Eq)(1 term) modus ponens ell d indeed
  [ meta fx ; meta m ] - [ meta fy ; meta m ] = 0 end line
line ell f because lemma sameNumerical modus ponens ell e indeed
  | [ meta fx ; meta m ] - [ meta fy ; meta m ] | = | 0 | end line
line ell g because lemma |0|=0 indeed | 0 | = 0 end line
line ell h because lemma eqTransitivity modus ponens ell f modus ponens ell g indeed 
  | [ meta fx ; meta m ] - [ meta fy ; meta m ] | = 0 end line

line ell i because lemma eqSymmetry modus ponens ell h indeed
  0 = | [ meta fx ; meta m ] - [ meta fy ; meta m ] | end line

because lemma subLessLeft modus ponens ell i modus ponens ell b indeed 
  | [ meta fx ; meta m ] - [ meta fy ; meta m ] | < meta ep end line
line ell big a end block

any term meta m comma meta n comma meta ep comma meta fx comma meta fy end line

because 1rule deduction modus ponens ell big a indeed
for all meta m indeed parenthesis meta n <= meta m imply [ meta fx ; meta m ] = [ meta fy ; meta m ] 
  end parenthesis imply
0 < meta ep imply meta n <= meta m imply
| [ meta fx ; meta m ] - [ meta fy ; meta m ] | < meta ep qed
end math ]"





"[  math in theory system Q lemma lemma eventually=f to sameF says
for all terms meta m comma meta n comma meta fx comma meta fy indeed
exist0 meta n indeed for all meta m indeed parenthesis 
  meta n <= meta m imply [ meta fx ; meta m ] = [ meta fy ; meta m ] end parenthesis infer
meta fx sameF meta fy
end lemma end math ]"

"[ math system Q proof of lemma eventually=f to sameF reads
any term meta m comma meta n comma meta fx comma meta fy end line
line ell a premise exist0 meta n indeed for all meta m indeed parenthesis 
  meta n <= meta m imply [ meta fx ; meta m ] = [ meta fy ; meta m ] end parenthesis end line
line ell b because lemma eventually=f to sameF helper indeed for all meta m indeed
  parenthesis meta n <= meta m imply [ meta fx ; meta m ] = [ meta fy ; meta m ] end parenthesis imply
  0 < meta ep imply meta n <= meta m imply
  | [ meta fx ; meta m ] - [ meta fy ; meta m ] | < meta ep end line
line ell c because pred lemma exist mp modus ponens ell b modus ponens ell a indeed
  0 < meta ep imply meta n <= meta m imply
  | [ meta fx ; meta m ] - [ meta fy ; meta m ] | < meta ep end line
line ell d because 1rule gen modus ponens ell c indeed
  for all meta m indeed parenthesis 0 < meta ep imply meta n <= meta m imply
  | [ meta fx ; meta m ] - [ meta fy ; meta m ] | < meta ep end parenthesis end line
line ell e because pred lemma intro exist at meta n modus ponens ell d indeed
  exist0 meta n indeed for all meta m indeed parenthesis 0 < meta ep imply meta n <= meta m imply
  | [ meta fx ; meta m ] - [ meta fy ; meta m ] | < meta ep end parenthesis end line
line ell f because 1rule gen modus ponens ell e indeed for all meta ep indeed
  exist0 meta n indeed for all meta m indeed parenthesis 0 < meta ep imply meta n <= meta m imply
  | [ meta fx ; meta m ] - [ meta fy ; meta m ] | < meta ep end parenthesis end line
line ell g because 1rule deduction modus ponens ell f indeed for all object ep indeed ""; XXded
  exist0 object n indeed for all object m indeed parenthesis 0 < object ep imply object n <= object m imply
  | [ meta fx ; object m ] - [ meta fy ; object m ] | < object ep end parenthesis end line

because 1rule repetition modus ponens ell g indeed meta fx sameF meta fy  qed ""; XXREP sameF
end math ]"

"[  math in theory system Q lemma lemma positiveToRight(Less) says
for all terms meta x comma meta y comma meta z indeed
meta x + meta y < meta z infer meta x < meta z - meta y
end lemma end math ]"

"[ math system Q proof of lemma positiveToRight(Less) reads
any term meta x comma meta y comma meta z end line
line ell a premise meta x + meta y < meta z end line
line ell b because lemma lessAddition modus ponens ell a indeed 
  meta x + meta y - meta y < meta z - meta y end line
line ell c because lemma x=x+y-y indeed meta x = meta x + meta y - meta y end line
line ell d because lemma eqSymmetry modus ponens ell c indeed
  meta x + meta y - meta y = meta x end line
because lemma subLessLeft modus ponens ell d modus ponens ell b indeed meta x < meta z - meta y qed
end math ]"


"[  math in theory system Q lemma lemma switchTerms(x-y<z) says
for all terms meta x comma meta y comma meta z indeed
meta x - meta y < meta z infer meta x - meta z < meta y 
end lemma end math ]"

"[ math system Q proof of lemma switchTerms(x-y<z) reads
any term meta x comma meta y comma meta z end line
line ell a premise meta x - meta y < meta z end line
line ell b because lemma negativeToRight(Less) modus ponens ell a indeed meta x < meta z + meta y end line
line ell c because axiom plusCommutativity indeed meta z + meta y = meta y + meta z end line
line ell d because lemma subLessRight modus ponens ell c modus ponens ell b indeed 
  meta x < meta y + meta z end line
because lemma positiveToLeft(Less) modus ponens ell d indeed meta x - meta z < meta y qed
end math ]"

"[  math in theory system Q lemma lemma switchTerms(x<y-z) says
for all terms meta x comma meta y comma meta z indeed
meta x < meta y - meta z infer meta z < meta y - meta x
end lemma end math ]"

"[ math system Q proof of lemma switchTerms(x<y-z) reads
any term meta x comma meta y comma meta z end line
line ell a premise meta x < meta y - meta z end line
line ell b because lemma negativeToLeft(Less) modus ponens ell a indeed meta x + meta z < meta y end line
line ell c because axiom plusCommutativity indeed meta x + meta z = meta z + meta x end line
line ell d because lemma subLessLeft modus ponens ell c modus ponens ell b indeed 
  meta z + meta x < meta y end line
because lemma positiveToRight(Less) modus ponens ell d indeed meta z < meta y - meta x qed
end math ]"


"[  math in theory system Q lemma lemma fromNot<f(Strong) helper2 says
for all terms meta x comma meta y comma meta z comma meta u comma meta v indeed
| meta x - meta y | < 1/3 * meta v infer
| meta z - meta u | < 1/3 * meta v infer
  meta y - meta u   < 1/3 * meta v infer meta x - meta z < meta v
end lemma end math ]"

"[ math system Q proof of lemma fromNot<f(Strong) helper2 reads
any term meta x comma meta y comma meta z comma meta u comma meta v end line
line ell a premise | meta x - meta y | < 1/3 * meta v end line
line ell b premise | meta z - meta u | < 1/3 * meta v end line
line ell c premise meta y - meta u     < 1/3 * meta v end line

line ell d because lemma x<=|x| indeed meta x - meta y <= | meta x - meta y | end line
line ell e because lemma leqLessTransitivity modus ponens ell d modus ponens ell a indeed
  meta x - meta y < 1/3 * meta v end line

line ell f because lemma numericalDifference indeed | meta z - meta u | = | meta u - meta z | end line
line ell g because lemma subLessLeft modus ponens ell f modus ponens ell b indeed 
  | meta u - meta z | < 1/3 * meta v end line
line ell h because lemma x<=|x| indeed meta u - meta z <= | meta u - meta z | end line
line ell i because lemma leqLessTransitivity modus ponens ell h modus ponens ell g indeed
  meta u - meta z < 1/3 * meta v end line

line ell j because lemma addEquations(Less) modus ponens ell e modus ponens ell c indeed
  parenthesis meta x - meta y end parenthesis + parenthesis meta y - meta u end parenthesis <
  1/3 * meta v + 1/3 * meta v end line
line ell k because lemma addEquations(Less) modus ponens ell j modus ponens ell i indeed
  parenthesis meta x - meta y end parenthesis + parenthesis meta y - meta u end parenthesis +
  parenthesis meta u - meta z end parenthesis < 1/3 * meta v + 1/3 * meta v + 1/3 * meta v end line

line ell l because lemma insertTwoMiddleTerms(Sum) indeed 
  meta x - meta z = parenthesis meta x - meta y end parenthesis + 
  parenthesis meta y - meta u end parenthesis + parenthesis meta u - meta z end parenthesis end line
line ell m because lemma eqSymmetry modus ponens ell l indeed
  parenthesis meta x - meta y end parenthesis + parenthesis meta y - meta u end parenthesis + 
  parenthesis meta u - meta z end parenthesis = meta x - meta z end line
line ell n because lemma subLessLeft modus ponens ell m modus ponens ell k indeed
  meta x - meta z < 1/3 * meta v + 1/3 * meta v + 1/3 * meta v end line

line ell o because lemma (1/3)x+(1/3)x+(1/3)x=x indeed 
  1/3 * meta v + 1/3 * meta v + 1/3 * meta v = meta v end line
because lemma subLessRight modus ponens ell o modus ponens ell n indeed meta x - meta z < meta v qed
end math ]"

"[  math in theory system Q lemma lemma fromNot<f(Strong) helper says
for all terms meta v1 comma meta v2 comma meta m comma meta n1 comma meta n2 comma meta ep 
                      comma meta fx comma meta fy indeed
parenthesis 0 < 1/3 * meta ep imply meta n1 <= meta n2 and0 
  [ meta fy ; meta n2 ] - 1/3 * meta ep < [ meta fx ; meta n2 ] end parenthesis imply
for all meta v1 comma meta v2 indeed parenthesis 0 < 1/3 * meta ep imply 
  meta n1 <= meta v1 imply meta n1 <= meta v2 imply
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < 1/3 * meta ep and0
  | [ meta fy ; meta v1 ] - [ meta fy ; meta v2 ] | < 1/3 * meta ep end parenthesis imply
0 < meta ep imply
meta n2 <= meta m imply
[ meta fy ; meta m ] - [ meta fx ; meta m ] < meta ep
end lemma end math ]"

"[  math system Q proof of lemma fromNot<f(Strong) helper reads
block any term meta v1 comma meta v2 comma meta m comma meta n1 comma meta n2 comma meta ep 
                       comma meta fx comma meta fy end line
line ell a premise 0 < 1/3 * meta ep imply meta n1 <= meta n2 and0 ""; ikke <f
  [ meta fy ; meta n2 ] - 1/3 * meta ep < [ meta fx ; meta n2 ] end line
line ell b premise for all meta v1 comma meta v2 indeed parenthesis 0 < 1/3 * meta ep imply ""; 2cauchy
  meta n1 <= meta v1 imply meta n1 <= meta v2 imply
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < 1/3 * meta ep and0
  | [ meta fy ; meta v1 ] - [ meta fy ; meta v2 ] | < 1/3 * meta ep end parenthesis end line
line ell c premise 0 < meta ep end line
line ell d premise meta n2 <= meta m end line

line ell big a because lemma 0<3 indeed 0 < 3 end line
line ell big b because lemma positiveInverted modus ponens ell big a indeed 0 < 1/3 end line
line ell big c because lemma positiveFactors modus ponens ell big b modus ponens ell c indeed
  0 < 1/3 * meta ep end line

line ell e because 1rule mp modus ponens ell a modus ponens ell big c indeed
  meta n1 <= meta n2 and0 [ meta fy ; meta n2 ] - 1/3 * meta ep < [ meta fx ; meta n2 ] end line
line ell f because prop lemma first conjunct modus ponens ell e indeed meta n1 <= meta n2 end line
line ell g because prop lemma second conjunct modus ponens ell e indeed 
  [ meta fy ; meta n2 ] - 1/3 * meta ep < [ meta fx ; meta n2 ] end line
line ell big d because lemma switchTerms(x-y<z) modus ponens ell g indeed
  [ meta fy ; meta n2 ] - [ meta fx ; meta n2 ] < 1/3 * meta ep end line

line ell h because lemma leqTransitivity modus ponens ell f modus ponens ell d indeed 
  meta n1 <= meta m end line

line ell i because lemma a4 at meta m modus ponens ell b indeed 
  for all meta v2 indeed parenthesis 0 < 1/3 * meta ep imply 
  meta n1 <= meta m imply meta n1 <= meta v2 imply
  | [ meta fx ; meta m ] - [ meta fx ; meta v2 ] | < 1/3 * meta ep and0
  | [ meta fy ; meta m ] - [ meta fy ; meta v2 ] | < 1/3 * meta ep end parenthesis end line
line ell j because lemma a4 at meta n2 modus ponens ell i indeed 
  0 < 1/3 * meta ep imply meta n1 <= meta m imply meta n1 <= meta n2 imply
  | [ meta fx ; meta m ] - [ meta fx ; meta n2 ] | < 1/3 * meta ep and0
  | [ meta fy ; meta m ] - [ meta fy ; meta n2 ] | < 1/3 * meta ep end line

line ell k because prop lemma mp3 
  modus ponens ell j modus ponens ell big c modus ponens ell h modus ponens ell f indeed
  | [ meta fx ; meta m ] - [ meta fx ; meta n2 ] | < 1/3 * meta ep and0
  | [ meta fy ; meta m ] - [ meta fy ; meta n2 ] | < 1/3 * meta ep end line

line ell l because prop lemma first conjunct modus ponens ell k indeed 
  | [ meta fx ; meta m ] - [ meta fx ; meta n2 ] | < 1/3 * meta ep end line
line ell m because prop lemma second conjunct modus ponens ell k indeed
  | [ meta fy ; meta m ] - [ meta fy ; meta n2 ] | < 1/3 * meta ep end line

because lemma fromNot<f(Strong) helper2 modus ponens ell m modus ponens ell l 
  modus ponens ell big d indeed [ meta fy ; meta m ] - [ meta fx ; meta m ] < meta ep end line
line ell big a end block

any term meta v1 comma meta v2 comma meta m comma meta n1 comma meta n2 comma meta ep 
                 comma meta fx comma meta fy end line 

because 1rule deduction modus ponens ell big a indeed
parenthesis 0 < 1/3 * meta ep imply meta n1 <= meta n2 and0 
  [ meta fy ; meta n2 ] - 1/3 * meta ep < [ meta fx ; meta n2 ] end parenthesis imply
for all meta v1 comma meta v2 indeed parenthesis 0 < 1/3 * meta ep imply 
  meta n1 <= meta v1 imply meta n1 <= meta v2 imply
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < 1/3 * meta ep and0
  | [ meta fy ; meta v1 ] - [ meta fy ; meta v2 ] | < 1/3 * meta ep end parenthesis imply
0 < meta ep imply
meta n2 <= meta m imply
[ meta fy ; meta m ] - [ meta fx ; meta m ] < meta ep qed
end math ]"

"[  math in theory system Q lemma lemma fromNot<f(Strong) says
""; XX ekstra variable
for all terms meta v1 comma meta v2 comma meta m comma meta n2 comma meta ep 
                      comma meta fx comma meta fy indeed
not0 meta fx <f meta fy infer
0 < meta ep imply meta n2 <= meta m imply [ meta fy ; meta m ] - [ meta fx ; meta m ] < meta ep
end lemma end math ]"

"[ math system Q proof of lemma fromNot<f(Strong) reads
any term meta v1 comma meta v2 comma meta m comma meta n2 comma meta ep comma meta fx comma meta fy end line
line ell a premise not0 meta fx <f meta fy end line
line ell b because lemma fromNot<f(Weak) modus ponens ell a indeed
  for all meta ep indeed for all meta n indeed exist0 meta n2 indeed parenthesis 0 < meta ep imply  
  meta n <= meta n2 and0 [ meta fy ; meta n2 ] - meta ep < [ meta fx ; meta n2 ] end parenthesis end line
line ell c because lemma a4 at 1/3 * meta ep modus ponens ell b indeed
  for all meta n indeed exist0 meta n2 indeed parenthesis 0 < 1/3 * meta ep imply  
  meta n <= meta n2 and0 [ meta fy ; meta n2 ] - 1/3 * meta ep < [ meta fx ; meta n2 ] end parenthesis 
  end line
line ell d because lemma a4 at meta n1 modus ponens ell c indeed 
  exist0 meta n2 indeed parenthesis 0 < 1/3 * meta ep imply  
  meta n1 <= meta n2 and0 [ meta fy ; meta n2 ] - 1/3 * meta ep < [ meta fx ; meta n2 ] end parenthesis 
  end line

line ell e because lemma 2cauchy indeed for all meta ep indeed exist0 meta n1 indeed
  for all meta v1 comma meta v2 indeed parenthesis 0 < meta ep imply 
  meta n1 <= meta v1 imply meta n1 <= meta v2 imply
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | <  meta ep and0
  | [ meta fy ; meta v1 ] - [ meta fy ; meta v2 ] | <  meta ep end parenthesis end line
line ell f because lemma a4 at 1/3 * meta ep modus ponens ell e indeed
  exist0 meta n1 indeed for all meta v1 comma meta v2 indeed parenthesis 0 < 1/3 * meta ep imply 
  meta n1 <= meta v1 imply meta n1 <= meta v2 imply
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | <  1/3 * meta ep and0
  | [ meta fy ; meta v1 ] - [ meta fy ; meta v2 ] | <  1/3 * meta ep end parenthesis end line

line ell g because lemma fromNot<f(Strong) helper indeed
parenthesis 0 < 1/3 * meta ep imply meta n1 <= meta n2 and0 
  [ meta fy ; meta n2 ] - 1/3 * meta ep < [ meta fx ; meta n2 ] end parenthesis imply
for all meta v1 comma meta v2 indeed parenthesis 0 < 1/3 * meta ep imply 
  meta n1 <= meta v1 imply meta n1 <= meta v2 imply
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < 1/3 * meta ep and0
  | [ meta fy ; meta v1 ] - [ meta fy ; meta v2 ] | < 1/3 * meta ep end parenthesis imply
0 < meta ep imply
meta n2 <= meta m imply
[ meta fy ; meta m ] - [ meta fx ; meta m ] < meta ep end line

because pred lemma exist mp2 modus ponens ell g modus ponens ell d modus ponens ell f indeed
  0 < meta ep imply meta n2 <= meta m imply [ meta fy ; meta m ] - [ meta fx ; meta m ] < meta ep qed
end math ]"

"[  math in theory system Q lemma lemma fromNotSameF(Strongest) helper2 says
for all terms meta x comma meta y comma meta z comma meta u comma meta v indeed
| meta x - meta y | < 1/3 * meta v     infer
| meta z - meta u | < 1/3 * meta v     infer
meta v <= | meta y - meta u | infer 1/3 * meta v < | meta x - meta z |
end lemma end math ]"

"[ math system Q proof of lemma fromNotSameF(Strongest) helper2 reads
any term meta x comma meta y comma meta z comma meta u comma meta v end line
line ell a premise | meta x - meta y | < 1/3 * meta v end line
line ell b premise | meta z - meta u | < 1/3 * meta v end line
line ell c premise meta v <= | meta y - meta u | end line

line ell big a because lemma numericalDifference indeed | meta x - meta y | = | meta y - meta x | end line 
line ell big b because lemma subLessLeft modus ponens ell big a modus ponens ell a indeed
  | meta y - meta x | < 1/3 * meta v end line
line ell d because lemma lessNegated modus ponens ell big b indeed 
  - parenthesis 1/3 * meta v end parenthesis < - | meta y - meta x | end line
line ell e because lemma lessNegated modus ponens ell b indeed 
  - parenthesis 1/3 * meta v end parenthesis < - | meta z - meta u | end line

line ell f because lemma addEquations(LeqLess) modus ponens ell c modus ponens ell d indeed
  meta v - parenthesis 1/3 * meta v end parenthesis < | meta y - meta u | - | meta y - meta x | end line
line ell g because lemma addEquations(Less) modus ponens ell f modus ponens ell e indeed
  meta v - parenthesis 1/3 * meta v end parenthesis - parenthesis 1/3 * meta v end parenthesis < 
  | meta y - meta u | - | meta y - meta x | - | meta z - meta u | end line


line ell h because lemma insertTwoMiddleTerms(Numerical) indeed 
  | meta y - meta u | <= | meta y - meta x | + | meta x - meta z | + | meta z - meta u | end line

line ell i because axiom plusAssociativity indeed
  | meta y - meta x | + | meta x - meta z | + | meta z - meta u | =
  | meta y - meta x | + parenthesis | meta x - meta z | + | meta z - meta u | end parenthesis end line
line ell j because axiom plusCommutativity indeed
  | meta y - meta x | + parenthesis | meta x - meta z | + | meta z - meta u | end parenthesis =
  | meta x - meta z | + | meta z - meta u | + | meta y - meta x | end line
line ell big j because lemma eqTransitivity modus ponens ell i modus ponens ell j indeed
  | meta y - meta x | + | meta x - meta z | + | meta z - meta u | =
  | meta x - meta z | + | meta z - meta u | + | meta y - meta x | end line
line ell k because lemma subLeqRight modus ponens ell big j modus ponens ell h indeed
  | meta y - meta u | <= | meta x - meta z | + | meta z - meta u | + | meta y - meta x | end line
line ell l because lemma positiveToLeft(Leq) modus ponens ell k indeed
  | meta y - meta u | - | meta y - meta x | <= | meta x - meta z | + | meta z - meta u | end line
line ell m because lemma positiveToLeft(Leq) modus ponens ell l indeed
  | meta y - meta u | - | meta y - meta x | - | meta z - meta u | <= | meta x - meta z | end line

line ell n because lemma lessLeqTransitivity modus ponens ell g modus ponens ell m indeed
  meta v - parenthesis 1/3 * meta v end parenthesis - parenthesis 1/3 * meta v end parenthesis < 
  | meta x - meta z | end line

line ell o because lemma (1/3)x+(1/3)x+(1/3)x=x indeed 
  1/3 * meta v + 1/3 * meta v + 1/3 * meta v = meta v end line
line ell p because lemma positiveToRight(Eq) modus ponens ell o indeed
  1/3 * meta v + 1/3 * meta v = meta v - 1/3 * meta v end line
line ell q because lemma positiveToRight(Eq) modus ponens ell p indeed
  1/3 * meta v = meta v - 1/3 * meta v - 1/3 * meta v end line
line ell s because lemma eqSymmetry modus ponens ell q indeed
  meta v - 1/3 * meta v - 1/3 * meta v = 1/3 * meta v end line

because lemma subLessLeft modus ponens ell s modus ponens ell n indeed 
  1/3 * meta v < | meta x - meta z | qed
end math ]"


"[ math in theory system Q lemma lemma fromNotSameF(Strongest) helper says
for all terms meta v1 comma meta v2 comma meta m comma meta n1 comma meta n2 comma meta ep 
                      comma meta fx comma meta fy indeed
not0 parenthesis 0 < meta ep imply meta n1 <= meta n2 imply 
  | [ meta fx ; meta n2 ] - [ meta fy ; meta n2 ] | < meta ep end parenthesis imply
for all meta v1 comma meta v2 indeed parenthesis 
  0 < 1/3 * meta ep imply meta n1 <= meta v1 imply meta n1 <= meta v2 imply 
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < 1/3 * meta ep and0
  | [ meta fy ; meta v1 ] - [ meta fy ; meta v2 ] | < 1/3 * meta ep end parenthesis imply
0 < 1/3 * meta ep and0 parenthesis meta n2 <= meta m imply 
  1/3 * meta ep < | [ meta fx ; meta m ] - [ meta fy ; meta m ] | end parenthesis
end lemma end math ]"



"[ math system Q proof of lemma fromNotSameF(Strongest) helper reads

block any term meta v1 comma meta v2 comma meta m comma meta n1 comma meta n2 comma meta ep 
                       comma meta fx comma meta fy end line

line ell a premise not0 parenthesis 0 < meta ep imply meta n1 <= meta n2 imply 
  | [ meta fx ; meta n2 ] - [ meta fy ; meta n2 ] | < meta ep end parenthesis end line
line ell b premise for all meta v1 comma meta v2 indeed parenthesis 
  0 < 1/3 * meta ep imply meta n1 <= meta v1 imply meta n1 <= meta v2 imply 
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < 1/3 * meta ep and0
  | [ meta fy ; meta v1 ] - [ meta fy ; meta v2 ] | < 1/3 * meta ep end parenthesis end line

line ell c premise meta n2 <= meta m end line

line ell d because prop lemma from negated double imply modus ponens ell a indeed
  0 < meta ep and0 meta n1 <= meta n2 and0 
  not0 | [ meta fx ; meta n2 ] - [ meta fy ; meta n2 ] | < meta ep end line

line ell e because prop lemma first conjunct modus ponens ell d indeed 
  0 < meta ep and0 meta n1 <= meta n2 end line
line ell f because prop lemma first conjunct modus ponens ell e indeed 0 < meta ep end line
line ell g because prop lemma second conjunct modus ponens ell e indeed meta n1 <= meta n2 end line
line ell h because prop lemma second conjunct modus ponens ell d indeed
  not0 | [ meta fx ; meta n2 ] - [ meta fy ; meta n2 ] | < meta ep end line
line ell i because lemma fromNotLess modus ponens ell h indeed
  meta ep <= | [ meta fx ; meta n2 ] - [ meta fy ; meta n2 ] | end line

line ell j because lemma a4 at meta m modus ponens ell b indeed
  for all meta v2 indeed parenthesis 
  0 < 1/3 * meta ep imply meta n1 <= meta m imply meta n1 <= meta v2 imply 
  | [ meta fx ; meta m ] - [ meta fx ; meta v2 ] | < 1/3 * meta ep and0
  | [ meta fy ; meta m ] - [ meta fy ; meta v2 ] | < 1/3 * meta ep end parenthesis end line
line ell k because lemma a4 at meta n2 modus ponens ell j indeed
  0 < 1/3 * meta ep imply meta n1 <= meta m imply meta n1 <= meta n2 imply 
  | [ meta fx ; meta m ] - [ meta fx ; meta n2 ] | < 1/3 * meta ep and0
  | [ meta fy ; meta m ] - [ meta fy ; meta n2 ] | < 1/3 * meta ep end line

line ell m because lemma 0<1/3 indeed 0 < 1/3 end line
line ell n because lemma positiveFactors modus ponens ell m modus ponens ell f indeed
  0 < 1/3 * meta ep end line

line ell o because lemma leqTransitivity modus ponens ell g modus ponens ell c indeed 
  meta n1 <= meta m end line

line ell p because prop lemma mp3 
  modus ponens ell k modus ponens ell n modus ponens ell o modus ponens ell g indeed
  | [ meta fx ; meta m ] - [ meta fx ; meta n2 ] | < 1/3 * meta ep and0
  | [ meta fy ; meta m ] - [ meta fy ; meta n2 ] | < 1/3 * meta ep end line
line ell q because prop lemma first conjunct modus ponens ell p indeed
  | [ meta fx ; meta m ] - [ meta fx ; meta n2 ] | < 1/3 * meta ep end line
line ell r because prop lemma second conjunct modus ponens ell p indeed
  | [ meta fy ; meta m ] - [ meta fy ; meta n2 ] | < 1/3 * meta ep end line

because lemma fromNotSameF(Strongest) helper2 modus ponens ell q modus ponens ell r
  modus ponens ell i indeed 1/3 * meta ep < | [ meta fx ; meta m ] - [ meta fy ; meta m ] | end line
line ell big x end block

block any term meta v1 comma meta v2 comma meta n1 comma meta n2 comma meta ep 
                       comma meta fx comma meta fy end line
line ell a premise not0 parenthesis 0 < meta ep imply meta n1 <= meta n2 imply 
  | [ meta fx ; meta n2 ] - [ meta fy ; meta n2 ] | < meta ep end parenthesis end line
line ell b premise for all meta v1 comma meta v2 indeed parenthesis 
  0 < 1/3 * meta ep imply meta n1 <= meta v1 imply meta n1 <= meta v2 imply 
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < 1/3 * meta ep and0
  | [ meta fy ; meta v1 ] - [ meta fy ; meta v2 ] | < 1/3 * meta ep end parenthesis end line
line ell c because prop lemma from negated imply modus ponens ell a indeed
  0 < meta ep and0 not0 parenthesis meta n1 <= meta n2 imply 
  | [ meta fx ; meta n2 ] - [ meta fy ; meta n2 ] | < meta ep end parenthesis end line
line ell d because prop lemma first conjunct modus ponens ell c indeed 0 < meta ep end line
line ell e because lemma 0<1/3 indeed 0 < 1/3 end line
because lemma positiveFactors modus ponens ell e modus ponens ell d indeed 0 < 1/3 * meta ep end line
line ell big q end block

any term meta v1 comma meta v2 comma meta m comma meta n1 comma meta n2 comma meta ep 
                  comma meta fx comma meta fy end line

line ell a because 1rule deduction modus ponens ell big x indeed
not0 parenthesis 0 < meta ep imply meta n1 <= meta n2 imply 
  | [ meta fx ; meta n2 ] - [ meta fy ; meta n2 ] | < meta ep end parenthesis imply
for all meta v1 comma meta v2 indeed parenthesis 
  0 < 1/3 * meta ep imply meta n1 <= meta v1 imply meta n1 <= meta v2 imply 
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < 1/3 * meta ep and0
  | [ meta fy ; meta v1 ] - [ meta fy ; meta v2 ] | < 1/3 * meta ep end parenthesis imply
meta n2 <= meta m imply 1/3 * meta ep < | [ meta fx ; meta m ] - [ meta fy ; meta m ] | end line

line ell b because 1rule deduction modus ponens ell big q indeed
not0 parenthesis 0 < meta ep imply meta n1 <= meta n2 imply 
  | [ meta fx ; meta n2 ] - [ meta fy ; meta n2 ] | < meta ep end parenthesis imply
for all meta v1 comma meta v2 indeed parenthesis 
  0 < 1/3 * meta ep imply meta n1 <= meta v1 imply meta n1 <= meta v2 imply 
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < 1/3 * meta ep and0
  | [ meta fy ; meta v1 ] - [ meta fy ; meta v2 ] | < 1/3 * meta ep end parenthesis imply
0 < 1/3 * meta ep end line

because prop lemma doubly conditioned join conjuncts modus ponens ell b modus ponens ell a indeed
not0 parenthesis 0 < meta ep imply meta n1 <= meta n2 imply 
  | [ meta fx ; meta n2 ] - [ meta fy ; meta n2 ] | < meta ep end parenthesis imply
for all meta v1 comma meta v2 indeed parenthesis 
  0 < 1/3 * meta ep imply meta n1 <= meta v1 imply meta n1 <= meta v2 imply 
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < 1/3 * meta ep and0
  | [ meta fy ; meta v1 ] - [ meta fy ; meta v2 ] | < 1/3 * meta ep end parenthesis imply
0 < 1/3 * meta ep and0 parenthesis meta n2 <= meta m imply 
  1/3 * meta ep < | [ meta fx ; meta m ] - [ meta fy ; meta m ] | end parenthesis qed
end math ]"






"[  math in theory system Q lemma lemma fromNotSameF(Strongest) says
""; XX ekstra variable
for all terms meta v1 comma meta v2 comma meta m comma meta n comma meta n1 comma meta n2 comma meta ep 
                      comma meta fx comma meta fy indeed
not0 meta fx sameF meta fy infer
exist0 meta ep indeed exist0 meta n1 indeed for all meta m indeed
  0 < meta ep and0 parenthesis meta n1 <= meta m imply 
  meta ep < | [ meta fx ; meta m ] - [ meta fy ; meta m ] | end parenthesis
end lemma end math ]"

"[ math system Q proof of lemma fromNotSameF(Strongest) reads
any term meta v1 comma meta v2 comma meta m comma meta n comma meta n1 comma meta n2 comma meta ep 
                 comma meta fx comma meta fy end line

line ell a premise not0 meta fx sameF meta fy end line
line ell x because 1rule repetition modus ponens ell a indeed ""; XXREP sameF
  not0 for all object ep indeed exist0 object n indeed for all object m indeed ( 
  0 < object ep imply object n <= object m imply 
  | [ meta fx ; object m ] - [ meta fy ; object m ] | < object ep ) end line
line ell b because 1rule deduction modus ponens ell x indeed ""; XXded
  not0 for all meta ep indeed exist0 meta n1 indeed for all meta n2 indeed parenthesis 
  0 < meta ep imply meta n1 <= meta n2 imply | [ meta fx ; meta n2 ] - [ meta fy ; meta n2 ] | < meta ep 
  end parenthesis end line
line ell c because pred lemma AEAnegated modus ponens ell b indeed
  exist0 meta ep indeed for all meta n1 indeed exist0 meta n2 indeed not0 parenthesis 
  0 < meta ep imply meta n1 <= meta n2 imply | [ meta fx ; meta n2 ] - [ meta fy ; meta n2 ] | < meta ep 
  end parenthesis end line
line ell d because lemma 2cauchy indeed
  for all meta ep indeed exist0 meta n1 indeed for all meta v1 comma meta v2 indeed parenthesis 
  0 < meta ep imply meta n1 <= meta v1 imply meta n1 <= meta v2 imply
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < meta ep and0
  | [ meta fy ; meta v1 ] - [ meta fy ; meta v2 ] | < meta ep end parenthesis end line
line ell e because lemma a4 at 1/3 * meta ep modus ponens ell d indeed
  exist0 meta n1 indeed for all meta v1 comma meta v2 indeed parenthesis 
  0 < 1/3 * meta ep imply meta n1 <= meta v1 imply meta n1 <= meta v2 imply
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < 1/3 * meta ep and0
  | [ meta fy ; meta v1 ] - [ meta fy ; meta v2 ] | < 1/3 * meta ep end parenthesis end line

line ell f because lemma fromNotSameF(Strongest) helper indeed
not0 parenthesis 0 < meta ep imply meta n1 <= meta n2 imply 
  | [ meta fx ; meta n2 ] - [ meta fy ; meta n2 ] | < meta ep end parenthesis imply
for all meta v1 comma meta v2 indeed parenthesis 
  0 < 1/3 * meta ep imply meta n1 <= meta v1 imply meta n1 <= meta v2 imply 
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < 1/3 * meta ep and0
  | [ meta fy ; meta v1 ] - [ meta fy ; meta v2 ] | < 1/3 * meta ep end parenthesis imply
0 < 1/3 * meta ep and0 parenthesis meta n2 <= meta m imply 
  1/3 * meta ep < | [ meta fx ; meta m ] - [ meta fy ; meta m ] | end parenthesis end line

line ell g because pred lemma EAE mp modus ponens ell f modus ponens ell c indeed
for all meta v1 comma meta v2 indeed parenthesis 
  0 < 1/3 * meta ep imply meta n1 <= meta v1 imply meta n1 <= meta v2 imply
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < 1/3 * meta ep and0
  | [ meta fy ; meta v1 ] - [ meta fy ; meta v2 ] | < 1/3 * meta ep end parenthesis imply
0 < 1/3 * meta ep and0 parenthesis meta n2 <= meta m imply 
  1/3 * meta ep < | [ meta fx ; meta m ] - [ meta fy ; meta m ] | end parenthesis end line

line ell h because pred lemma exist mp modus ponens ell g modus ponens ell e indeed 
  0 < 1/3 * meta ep and0 parenthesis meta n2 <= meta m imply 
  1/3 * meta ep < | [ meta fx ; meta m ] - [ meta fy ; meta m ] | end parenthesis end line

line ell i because 1rule gen modus ponens ell h indeed for all meta m indeed
  0 < 1/3 * meta ep and0 parenthesis meta n2 <= meta m imply 
  1/3 * meta ep < | [ meta fx ; meta m ] - [ meta fy ; meta m ] | end parenthesis end line

line ell j because pred lemma intro exist at meta n2 modus ponens ell i indeed 
  exist0 meta n indeed for all meta m indeed
  0 < 1/3 * meta ep and0 parenthesis meta n1 <= meta m imply 
  1/3 * meta ep < | [ meta fx ; meta m ] - [ meta fy ; meta m ] | end parenthesis end line

because pred lemma intro exist at 1/3 * meta ep modus ponens ell j indeed 
  exist0 meta ep indeed exist0 meta n1 indeed for all meta m indeed
  0 < meta ep and0 parenthesis meta n1 <= meta m imply 
  meta ep < | [ meta fx ; meta m ] - [ meta fy ; meta m ] | end parenthesis qed
end math ]"

"[  math in theory system Q lemma lemma toLess(F) helper says
for all terms meta m comma meta n1 comma meta n2 comma meta ep comma meta fx comma meta fy indeed
for all meta m indeed 0 < meta ep and0 parenthesis meta n1 <= meta m imply 
  meta ep < | [ meta fx ; meta m ] - [ meta fy ; meta m ] | end parenthesis imply
parenthesis 0 < meta ep imply  
  meta n2 <= meta m imply [ meta fy ; meta m ] - [ meta fx ; meta m ] < meta ep end parenthesis imply
0 < meta ep and0 parenthesis max( meta n1 , meta n2 ) <= meta m imply 
  [ meta fy ; meta m ] <= [ meta fx ; meta m ] - meta ep end parenthesis
end lemma end math ]"

"[ math system Q proof of lemma toLess(F) helper reads
block any term meta m comma meta n1 comma meta n2 comma meta ep comma meta fx comma meta fy end line
line ell a premise 
  "";exist0 meta ep indeed exist0 meta n indeed 
  for all meta m indeed 0 < meta ep and0 parenthesis meta n1 <= meta m imply 
  meta ep < | [ meta fx ; meta m ] - [ meta fy ; meta m ] | end parenthesis end line
line ell b premise ""; fromNot<f(strong)
  0 < meta ep imply meta n2 <= meta m imply [ meta fy ; meta m ] - [ meta fx ; meta m ] < meta ep end line

line ell c premise max( meta n1 , meta n2 ) <= meta m end line
line ell d because lemma leqMax1 indeed meta n1 <= max( meta n1 , meta n2 ) end line
line ell e because lemma leqTransitivity modus ponens ell d modus ponens ell c indeed 
  meta n1 <= meta m end line

line ell g because lemma leqMax2 indeed meta n2 <= max( meta n1 , meta n2 ) end line
line ell h because lemma leqTransitivity modus ponens ell g modus ponens ell c indeed
  meta n2 <= meta m end line

line ell i because lemma a4 at meta m modus ponens ell a indeed
  0 < meta ep and0 parenthesis meta n1 <= meta m imply 
  meta ep < | [ meta fx ; meta m ] - [ meta fy ; meta m ] | end parenthesis end line
line ell j because prop lemma first conjunct modus ponens ell i indeed 0 < meta ep end line
line ell k because prop lemma second conjunct modus ponens ell i indeed
  meta n1 <= meta m imply meta ep < | [ meta fx ; meta m ] - [ meta fy ; meta m ] | end line
line ell l because 1rule mp modus ponens ell k modus ponens ell e indeed
  meta ep < | [ meta fx ; meta m ] - [ meta fy ; meta m ] | end line
line ell m because lemma fromNumericalGreater modus ponens ell l indeed
  [ meta fx ; meta m ] - [ meta fy ; meta m ] < - meta ep or0 
  meta ep < [ meta fx ; meta m ] - [ meta fy ; meta m ] end line



line ell n because prop lemma mp2 modus ponens ell b modus ponens ell j modus ponens ell h indeed
  [ meta fy ; meta m ] - [ meta fx ; meta m ] < meta ep end line
line ell o because lemma lessNegated modus ponens ell n indeed
  - meta ep < - parenthesis [ meta fy ; meta m ] - [ meta fx ; meta m ] end parenthesis end line
line ell p because lemma minusNegated indeed
  - parenthesis [ meta fy ; meta m ] - [ meta fx ; meta m ] end parenthesis =
  [ meta fx ; meta m ] - [ meta fy ; meta m ] end line
line ell q because lemma subLessRight modus ponens ell p modus ponens ell o indeed
  - meta ep < [ meta fx ; meta m ] - [ meta fy ; meta m ] end line

line ell r because lemma lessLeq modus ponens ell q indeed
  - meta ep <= [ meta fx ; meta m ] - [ meta fy ; meta m ] end line
line ell s because lemma toNotLess modus ponens ell r indeed
  not0 [ meta fx ; meta m ] - [ meta fy ; meta m ] < - meta ep end line
line ell t because prop lemma negate first disjunct modus ponens ell m modus ponens ell s indeed
  meta ep < [ meta fx ; meta m ] - [ meta fy ; meta m ] end line
line ell u because lemma switchTerms(x<y-z) modus ponens ell t indeed
  [ meta fy ; meta m ] < [ meta fx ; meta m ] - meta ep end line
because lemma lessLeq modus ponens ell u indeed 
  [ meta fy ; meta m ] <= [ meta fx ; meta m ] - meta ep end line
line ell big a end block

block any term meta m comma meta n1 comma meta n2 comma meta ep comma meta fx comma meta fy end line
line ell a premise 
  "";exist0 meta ep indeed exist0 meta n indeed 
  for all meta m indeed 0 < meta ep and0 parenthesis meta n1 <= meta m imply 
  meta ep < | [ meta fx ; meta m ] - [ meta fy ; meta m ] | end parenthesis end line

line ell b premise  ""; bruges ikke, skal bare vaere der
  0 < meta ep imply meta n2 <= meta m imply [ meta fy ; meta m ] - [ meta fx ; meta m ] < meta ep end line

line ell c because lemma a4 at meta m modus ponens ell a indeed
  0 < meta ep and0 parenthesis meta n1 <= meta m imply 
  meta ep <= | [ meta fx ; meta m ] - [ meta fy ; meta m ] | end parenthesis end line
because prop lemma first conjunct modus ponens ell c indeed 0 < meta ep end line
line ell big b end block


any term meta m comma meta n1 comma meta n2 comma meta ep comma meta fx comma meta fy end line 

line ell a because 1rule deduction modus ponens ell big a indeed
for all meta m indeed 0 < meta ep and0 parenthesis meta n1 <= meta m imply 
  meta ep < | [ meta fx ; meta m ] - [ meta fy ; meta m ] | end parenthesis imply
parenthesis 0 < meta ep imply  
  meta n2 <= meta m imply [ meta fy ; meta m ] - [ meta fx ; meta m ] < meta ep end parenthesis imply
max( meta n1 , meta n2 ) <= meta m imply [ meta fy ; meta m ] <= [ meta fx ; meta m ] - meta ep end line

line ell b because 1rule deduction modus ponens ell big b indeed 
for all meta m indeed 0 < meta ep and0 parenthesis meta n1 <= meta m imply 
  meta ep < | [ meta fx ; meta m ] - [ meta fy ; meta m ] | end parenthesis imply

parenthesis 0 < meta ep imply  
  meta n2 <= meta m imply [ meta fy ; meta m ] - [ meta fx ; meta m ] < meta ep end parenthesis imply
0 < meta ep end line

because prop lemma doubly conditioned join conjuncts modus ponens ell b modus ponens ell a indeed
for all meta m indeed 0 < meta ep and0 parenthesis meta n1 <= meta m imply 
  meta ep < | [ meta fx ; meta m ] - [ meta fy ; meta m ] | end parenthesis imply
parenthesis 0 < meta ep imply  
  meta n2 <= meta m imply [ meta fy ; meta m ] - [ meta fx ; meta m ] < meta ep end parenthesis imply
0 < meta ep and0 parenthesis max( meta n1 , meta n2 ) <= meta m imply 
  [ meta fy ; meta m ] <= [ meta fx ; meta m ] - meta ep end parenthesis qed
end math ]"




"[  math in theory system Q lemma lemma toLess(F) says
""; ekstra variable
for all terms meta m comma meta n comma meta n1 comma meta n2 comma meta ep 
                     comma meta fx comma meta fy indeed
not0 meta fx <=f meta fy infer meta fy <f meta fx
end lemma end math ]"

"[ math system Q proof of lemma toLess(F) reads
any term meta m comma meta n comma meta n1 comma meta n2 comma meta ep 
                comma meta fx comma meta fy end line
line ell a premise not0 meta fx <=f meta fy end line
line ell b because 1rule repetition modus ponens ell a indeed ""; XXREP <=f
  not0 parenthesis meta fx <f meta fy or0 meta fx sameF meta fy end parenthesis end line
line ell c because prop lemma from negated or modus ponens ell b indeed
  not0 meta fx <f meta fy and0 not0 meta fx sameF meta fy end line

line ell d because prop lemma first conjunct modus ponens ell c indeed
  not0 meta fx <f meta fy end line
line ell e because prop lemma second conjunct modus ponens ell c indeed
  not0 meta fx sameF meta fy end line

line ell f because lemma fromNot<f(Strong) modus ponens ell d indeed
  0 < meta ep imply meta n2 <= meta m imply [ meta fy ; meta m ] - [ meta fx ; meta m ] < meta ep end line

line ell g because lemma fromNotSameF(Strongest) modus ponens ell e indeed
  exist0 meta ep indeed exist0 meta n1 indeed for all meta m indeed 
  0 < meta ep and0 parenthesis meta n1 <= meta m imply 
  meta ep < | [ meta fx ; meta m ] - [ meta fy ; meta m ] | end parenthesis end line


line ell h because lemma toLess(F) helper indeed
for all meta m indeed 0 < meta ep and0 parenthesis meta n1 <= meta m imply 
  meta ep < | [ meta fx ; meta m ] - [ meta fy ; meta m ] | end parenthesis imply
parenthesis 0 < meta ep imply  
  meta n2 <= meta m imply [ meta fy ; meta m ] - [ meta fx ; meta m ] < meta ep end parenthesis imply
0 < meta ep and0 parenthesis max( meta n1 , meta n2 ) <= meta m imply 
  [ meta fy ; meta m ] <= [ meta fx ; meta m ] - meta ep end parenthesis end line



line ell i because pred lemma 2exist mp modus ponens ell h modus ponens ell g indeed
parenthesis 0 < meta ep imply  
  meta n2 <= meta m imply [ meta fy ; meta m ] - [ meta fx ; meta m ] < meta ep end parenthesis imply
0 < meta ep and0 parenthesis max( meta n1 , meta n2 ) <= meta m imply 
  [ meta fy ; meta m ] <= [ meta fx ; meta m ] - meta ep end parenthesis end line

line ell j because 1rule mp modus ponens ell i modus ponens ell f indeed
  0 < meta ep and0 parenthesis max( meta n1 , meta n2 ) <= meta m imply 
  [ meta fy ; meta m ] <= [ meta fx ; meta m ] - meta ep end parenthesis end line

line ell k because 1rule gen modus ponens ell j indeed for all meta m indeed parenthesis
  0 < meta ep and0 parenthesis max( meta n1 , meta n2 ) <= meta m imply 
  [ meta fy ; meta m ] <= [ meta fx ; meta m ] - meta ep end parenthesis end parenthesis end line
line ell l because pred lemma intro exist at max( meta n1 , meta n2 ) modus ponens ell k indeed 
  exist0 meta n indeed for all meta m indeed parenthesis
  0 < meta ep and0 parenthesis meta n <= meta m imply 
  [ meta fy ; meta m ] <= [ meta fx ; meta m ] - meta ep end parenthesis end parenthesis end line
line ell m because pred lemma intro exist at meta ep modus ponens ell l indeed 
  exist0 meta ep indeed exist0 meta n indeed for all meta m indeed parenthesis
  0 < meta ep and0 parenthesis meta n <= meta m imply 
  [ meta fy ; meta m ] <= [ meta fx ; meta m ] - meta ep end parenthesis end parenthesis end line
line ell x because 1rule deduction modus ponens ell m indeed ""; XXded
  exist0 object ep indeed exist0 object n indeed for all object m indeed (
  0 < object ep and0 ( object n <= object m imply 
  [ meta fy ; object m ] <= [ meta fx ; object m ] - object ep ) ) end line
because 1rule repetition modus ponens ell x indeed meta fy <f meta fx qed ""; XXREP <f
end math ]"


"[  math in theory system Q lemma lemma from!!== says
for all terms meta m comma meta n comma meta ep comma meta fx comma meta fy indeed 
""; XX lidt grimt med de ekstra variable
R( meta fx ) !!== R( meta fy ) infer not0 meta fx sameF meta fy
end lemma end math ]"

"[ math system Q proof of lemma from!!== reads

block any term meta m comma meta n comma meta ep comma meta fx comma meta fy end line
line ell a premise meta fx sameF meta fy end line
because 1rule to== modus ponens ell a indeed R( meta fx ) == R( meta fy ) end line
line ell big a end block

any term meta m comma meta n comma meta ep comma meta fx comma meta fy end line

line ell a because 1rule deduction modus ponens ell big a indeed
  meta fx sameF meta fy imply R( meta fx ) == R( meta fy ) end line

line ell b premise R( meta fx ) !!== R( meta fy ) end line

because prop lemma mt modus ponens ell a modus ponens ell b indeed not0 meta fx sameF meta fy qed
end math ]"

"[  math in theory system Q lemma lemma toLess(R) says
for all terms meta fx comma meta fy indeed
not0 R( meta fx ) <<== R( meta fy ) infer R( meta fy ) << R( meta fx )
end lemma end math ]"

"[ math system Q proof of lemma toLess(R) reads
any term meta fx comma meta fy end line
line ell a premise not0 R( meta fx ) <<== R( meta fy ) end line
line ell b because 1rule repetition modus ponens ell a indeed ""; XXREP <<==
  not0 parenthesis R( meta fx ) << R( meta fy ) or0 R( meta fx ) == R( meta fy ) end parenthesis end line
line ell c because prop lemma from negated or modus ponens ell b indeed
  not0 R( meta fx ) << R( meta fy ) and0 R( meta fx ) !!== R( meta fy ) end line  

line ell d because prop lemma first conjunct modus ponens ell c indeed 
  not0 R( meta fx ) << R( meta fy ) end line
line ell e because prop lemma second conjunct modus ponens ell c indeed
  R( meta fx ) !!== R( meta fy ) end line

line ell f because lemma fromNot<< modus ponens ell d indeed
  not0 meta fx <f meta fy end line  
line ell g because lemma from!!== modus ponens ell e indeed
  not0 meta fx sameF meta fy end line
line ell h because prop lemma join conjuncts modus ponens ell f modus ponens ell g indeed
  not0 meta fx <f meta fy and0 not0 meta fx sameF meta fy end line
line ell i because prop lemma to negated or modus ponens ell h indeed
  not0 parenthesis meta fx <f meta fy or0 meta fx sameF meta fy end parenthesis end line
line ell j because 1rule repetition modus ponens ell i indeed ""; XXREP <=f
  not0 meta fx <=f meta fy end line

line ell k because lemma toLess(F) modus ponens ell j indeed meta fy <f meta fx end line
  
because 1rule repetition modus ponens ell k indeed R( meta fy ) << R( meta fx ) qed ""; XXREP <<
end math ]"

"[  math in theory system Q lemma lemma fromNotLess(R) says
for all terms meta fx comma meta fy indeed
not0 R( meta fx ) << R( meta fy ) infer R( meta fy ) <<== R( meta fx )
end lemma end math ]"

"[ math system Q proof of lemma fromNotLess(R) reads
block any term meta fx comma meta fy end line
line ell a premise not0 R( meta fy ) <<== R( meta fx ) end line
because lemma toLess(R) modus ponens ell a indeed R( meta fx ) << R( meta fy ) end line
line ell big a end block

any term meta fx comma meta fy end line
line ell a because 1rule deduction modus ponens ell big a indeed
  not0 R( meta fy ) <<== R( meta fx ) imply R( meta fx ) << R( meta fy ) end line
  
line ell b premise not0 R( meta fx ) << R( meta fy ) end line

because prop lemma negative mt modus ponens ell a modus ponens ell b indeed 
  R( meta fy ) <<== R( meta fx ) qed
end math ]"


"[  math in theory system Q lemma lemma positiveToLeft(Leq) says
for all terms meta x comma meta y comma meta z indeed
meta x <= meta y + meta z infer meta x - meta z <= meta y
end lemma end math ]"

"[ math system Q proof of lemma positiveToLeft(Leq) reads
any term meta x comma meta y comma meta z end line
line ell a premise meta x <= meta y + meta z end line

line ell b because lemma leqAddition modus ponens ell a indeed 
  meta x - meta z <= meta y + meta z - meta z end line
line ell c because lemma x=x+y-y indeed meta y = meta y + meta z - meta z end line
line ell d because lemma eqSymmetry modus ponens ell c indeed meta y + meta z - meta z = meta y end line

because lemma subLeqRight modus ponens ell d modus ponens ell b indeed meta x - meta z <= meta y qed
end math ]"



"[  math in theory system Q lemma lemma fromNotSameF(Strong) helper2 says
for all terms meta x comma meta y comma meta z comma meta u comma meta v indeed
meta v <= | meta x - meta z | infer
| meta x - meta y | < 1/2 * meta v infer | meta z - meta u | < 1/2 * meta v infer
meta y != meta u  
end lemma end math ]"

"[ math system Q proof of lemma fromNotSameF(Strong) helper2 reads
any term meta x comma meta y comma meta z comma meta u comma meta v end line
line ell c premise meta v <= | meta x - meta z | end line
line ell a premise | meta x - meta y | < 1/2 * meta v end line
line ell b premise | meta z - meta u | < 1/2 * meta v end line

line ell d because lemma lessNegated modus ponens ell a indeed 
  - parenthesis 1/2 * meta v end parenthesis < - | meta x - meta y | end line 

line ell big a because lemma numericalDifference indeed | meta z - meta u | = | meta u - meta z | end line
line ell big b because lemma subLessLeft modus ponens ell big a modus ponens ell b indeed
  | meta u - meta z | < 1/2 * meta v end line

line ell e because lemma lessNegated modus ponens ell big b indeed
  - parenthesis 1/2 * meta v end parenthesis < - | meta u - meta z | end line
line ell f because lemma addEquations(Less) modus ponens ell d modus ponens ell e indeed
  - parenthesis 1/2 * meta v end parenthesis - parenthesis 1/2 * meta v end parenthesis < 
  - | meta x - meta y | - | meta u - meta z | end line

line ell g because lemma (1/2)x+(1/2)x=x indeed 1/2 * meta v + 1/2 * meta v = meta v end line
line ell h because lemma eqNegated modus ponens ell g indeed 
  - parenthesis 1/2 * meta v + 1/2 * meta v end parenthesis = - meta v end line

line ell i because lemma -x-y=-(x+y) indeed 
  - parenthesis 1/2 * meta v end parenthesis - parenthesis 1/2 * meta v end parenthesis =
  - parenthesis 1/2 * meta v + 1/2 * meta v end parenthesis end line
line ell j because lemma eqTransitivity modus ponens ell i modus ponens ell h indeed
  - parenthesis 1/2 * meta v end parenthesis - parenthesis 1/2 * meta v end parenthesis = - meta v end line

line ell k because lemma subLessLeft modus ponens ell j modus ponens ell f indeed
  - meta v < - | meta x - meta y | - | meta u - meta z | end line

line ell l because axiom plusCommutativity indeed   
  - | meta x - meta y | - | meta u - meta z | = - | meta u - meta z | - | meta x - meta y | end line
line ell m because lemma subLessRight modus ponens ell l modus ponens ell k indeed
  - meta v <  - | meta u - meta z | - | meta x - meta y | end line

line ell n because lemma addEquations(LeqLess) modus ponens ell c modus ponens ell m indeed
  meta v - meta v < | meta x - meta z | + 
  parenthesis - | meta u - meta z | - | meta x - meta y | end parenthesis end line

line ell o because axiom negative indeed meta v - meta v = 0 end line
line ell p because lemma subLessLeft modus ponens ell o modus ponens ell n indeed
  0 < | meta x - meta z | + parenthesis - | meta u - meta z | - | meta x - meta y | end parenthesis end line

line ell q because axiom plusAssociativity indeed 
  | meta x - meta z | - | meta u - meta z | - | meta x - meta y | = 
  | meta x - meta z | + parenthesis - | meta u - meta z | - | meta x - meta y | end parenthesis end line
line ell r because lemma eqSymmetry modus ponens ell q indeed
  | meta x - meta z | + parenthesis - | meta u - meta z | - | meta x - meta y | end parenthesis =
  | meta x - meta z | - | meta u - meta z | - | meta x - meta y | end line
line ell s  because lemma subLessRight modus ponens ell r modus ponens ell p indeed
  0 < | meta x - meta z | - | meta u - meta z | - | meta x - meta y | end line

line ell t because lemma insertTwoMiddleTerms(Numerical) indeed
  | meta x - meta z | <= | meta x - meta y | + | meta y - meta u | + | meta u - meta z | end line

line ell u because lemma positiveToLeft(Leq) modus ponens ell t indeed
  | meta x - meta z | - | meta u - meta z | <= | meta x - meta y | + | meta y - meta u | end line

line ell v because axiom plusCommutativity indeed 
  | meta x - meta y | + | meta y - meta u | = | meta y - meta u | + | meta x - meta y | end line
line ell x because lemma subLeqRight modus ponens ell v modus ponens ell u indeed
  | meta x - meta z | - | meta u - meta z | <= | meta y - meta u | + | meta x - meta y | end line

line ell y because lemma positiveToLeft(Leq) modus ponens ell x indeed
  | meta x - meta z | - | meta u - meta z | - | meta x - meta y | <= | meta y - meta u | end line

line ell z because lemma lessLeqTransitivity modus ponens ell s modus ponens ell y indeed
  0 < | meta y - meta u | end line

line ell big a because lemma fromPositiveNumerical modus ponens ell z indeed
  meta y - meta u != 0 end line

because lemma negativeToRight(Neq)(1 term) modus ponens ell big a indeed meta y != meta u qed
end math ]"


"[  math in theory system Q lemma lemma fromNotSameF(Strong) helper says
for all terms meta v1 comma meta v2 comma meta m comma meta n1 comma meta n2 
                      comma meta ep comma meta fx comma meta fy indeed 
not0 parenthesis 
  0 < meta ep imply meta n1 <= meta n2 imply | [ meta fx ; meta n2 ] - [ meta fy ; meta n2 ] | < meta ep
  end parenthesis imply
for all meta v1 comma meta v2 indeed parenthesis 
  0 < 1/2 * meta ep imply meta n1 <= meta v1 imply meta n1 <= meta v2 imply 
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < 1/2 * meta ep and0
  | [ meta fy ; meta v1 ] - [ meta fy ; meta v2 ] | < 1/2 * meta ep end parenthesis imply
meta n2 <= meta m imply
[ meta fx ; meta m ] !=  [ meta fy ; meta m ]
end lemma end math ]"


"[ math system Q proof of lemma fromNotSameF(Strong) helper reads

block any term meta v1 comma meta v2 comma meta m comma meta n1 comma meta n2 
                       comma meta ep comma meta fx comma meta fy end line

line ell b premise not0 parenthesis 
  0 < meta ep imply meta n1 <= meta n2 imply | [ meta fx ; meta n2 ] - [ meta fy ; meta n2 ] | < meta ep
  end parenthesis end line

line ell a premise for all meta v1 comma meta v2 indeed parenthesis 
  0 < 1/2 * meta ep imply meta n1 <= meta v1 imply meta n1 <= meta v2 imply 
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < 1/2 * meta ep and0
  | [ meta fy ; meta v1 ] - [ meta fy ; meta v2 ] | < 1/2 * meta ep end parenthesis end line

line ell c premise meta n2 <= meta m end line

line ell d because prop lemma from negated double imply modus ponens ell b indeed 0 < meta ep and0
  meta n1 <= meta n2 and0 not0 | [ meta fx ; meta n2 ] - [ meta fy ; meta n2 ] | < meta ep 
  end line
line ell e because prop lemma first conjunct modus ponens ell d indeed 
  0 < meta ep and0 meta n1 <= meta n2 end line
line ell f because prop lemma first conjunct modus ponens ell e indeed 0 < meta ep end line
line ell g because prop lemma second conjunct modus ponens ell e indeed meta n1 <= meta n2 end line
line ell h because prop lemma second conjunct modus ponens ell d indeed
  not0 | [ meta fx ; meta n2 ] - [ meta fy ; meta n2 ] | < meta ep end line


line ell big i because lemma fromNotLess modus ponens ell h indeed
  meta ep <= | [ meta fx ; meta n2 ] - [ meta fy ; meta n2 ] | end line

line ell i because lemma a4 at meta n2 modus ponens ell a indeed
  for all meta v2 indeed parenthesis 
  0 < 1/2 * meta ep imply meta n1 <= meta n2 imply meta n1 <= meta v2 imply 
  | [ meta fx ; meta n2 ] - [ meta fx ; meta v2 ] | < 1/2 * meta ep and0
  | [ meta fy ; meta n2 ] - [ meta fy ; meta v2 ] | < 1/2 * meta ep end parenthesis end line
line ell j because lemma a4 at meta m modus ponens ell i indeed
  0 < 1/2 * meta ep imply meta n1 <= meta n2 imply meta n1 <= meta m imply 
  | [ meta fx ; meta n2 ] - [ meta fx ; meta m ] | < 1/2 * meta ep and0
  | [ meta fy ; meta n2 ] - [ meta fy ; meta m ] | < 1/2 * meta ep end line

line ell k because lemma positiveHalved modus ponens ell f indeed 0 < 1/2 * meta ep end line
line ell m because lemma leqTransitivity modus ponens ell g modus ponens ell c indeed
  meta n1 <= meta m end line
line ell n because prop lemma mp3 modus ponens ell j modus ponens ell k modus ponens ell g 
  modus ponens ell m indeed | [ meta fx ; meta n2 ] - [ meta fx ; meta m ] | < 1/2 * meta ep and0 
  | [ meta fy ; meta n2 ] - [ meta fy ; meta m ] | < 1/2 * meta ep end line
line ell o because prop lemma first conjunct modus ponens ell n indeed
  | [ meta fx ; meta n2 ] - [ meta fx ; meta m ] | < 1/2 * meta ep end line
line ell p because prop lemma second conjunct modus ponens ell n indeed
  | [ meta fy ; meta n2 ] - [ meta fy ; meta m ] | < 1/2 * meta ep end line

because lemma fromNotSameF(Strong) helper2 modus ponens ell big i modus ponens ell o 
  modus ponens ell p indeed [ meta fx ; meta m ] != [ meta fy ; meta m ] end line
line ell big a end block

any term meta v1 comma meta v2 comma meta m comma meta n1 comma meta n2 
                 comma meta ep comma meta fx comma meta fy end line

because 1rule deduction modus ponens ell big a indeed  
not0 parenthesis 
  0 < meta ep imply meta n1 <= meta n2 imply | [ meta fx ; meta n2 ] - [ meta fy ; meta n2 ] | < meta ep
  end parenthesis imply
for all meta v1 comma meta v2 indeed parenthesis 
  0 < 1/2 * meta ep imply meta n1 <= meta v1 imply meta n1 <= meta v2 imply 
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < 1/2 * meta ep and0
  | [ meta fy ; meta v1 ] - [ meta fy ; meta v2 ] | < 1/2 * meta ep end parenthesis imply
meta n2 <= meta m imply
[ meta fx ; meta m ] !=  [ meta fy ; meta m ] qed
end math ]"




"[  math in theory system Q lemma lemma fromNotSameF(Strong) says
""; XX lidt grimt med de ekstra variable
for all terms meta v1 comma meta v2 comma meta m comma meta n comma meta n1 comma meta n2 comma meta ep 
                      comma meta fx comma meta fy indeed
not0 meta fx sameF meta fy infer
exist0 meta n indeed for all meta m indeed parenthesis
  meta n <= meta m imply  
  [ meta fx ; meta m ] != [ meta fy ; meta m ] end parenthesis
end lemma end math ]"


"[ math system Q proof of lemma fromNotSameF(Strong) reads
any term meta v1 comma meta v2 comma meta m comma meta n comma meta n1 comma meta n2 comma meta ep 
                 comma meta fx comma meta fy end line
line ell a premise not0 meta fx sameF meta fy end line
line ell x because 1rule repetition modus ponens ell a indeed  ""; XXREP sameF
  not0 for all object ep indeed exist0 object n indeed for all object m indeed ( 
  0 < object ep imply object n <= object m imply 
  | [ meta fx ; object m ] - [ meta fy ; object m ] | < object ep ) end line
line ell b because 1rule deduction modus ponens ell x indeed ""; XXded
  not0 for all meta ep indeed exist0 meta n1 indeed for all meta n2 indeed parenthesis 
  0 < meta ep imply meta n1 <= meta n2 imply | [ meta fx ; meta n2 ] - [ meta fy ; meta n2 ] | < meta ep 
  end parenthesis end line
line ell c because pred lemma AEAnegated modus ponens ell b indeed
  exist0 meta ep indeed for all meta n1 indeed exist0 meta n2 indeed not0 parenthesis 
  0 < meta ep imply meta n1 <= meta n2 imply | [ meta fx ; meta n2 ] - [ meta fy ; meta n2 ] | < meta ep 
  end parenthesis end line

line ell d because lemma 2cauchy indeed for all meta ep indeed exist0 meta n1 indeed 
  for all meta v1 comma meta v2 indeed parenthesis 0 < meta ep imply 
  meta n1 <= meta v1 imply meta n1 <= meta v2 imply
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < meta ep and0
  | [ meta fy ; meta v1 ] - [ meta fy ; meta v2 ] | < meta ep end parenthesis end line

line ell e because lemma a4 at 1/2 * meta ep modus ponens ell d indeed exist0 meta n1 indeed 
  for all meta v1 comma meta v2 indeed parenthesis 0 < 1/2 * meta ep imply
  meta n1 <= meta v1 imply meta n1 <= meta v2 imply
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < 1/2 * meta ep and0
  | [ meta fy ; meta v1 ] - [ meta fy ; meta v2 ] | < 1/2 * meta ep end parenthesis end line

line ell f because lemma fromNotSameF(Strong) helper indeed
not0 parenthesis 
  0 < meta ep imply meta n1 <= meta n2 imply | [ meta fx ; meta n2 ] - [ meta fy ; meta n2 ] | < meta ep
  end parenthesis imply
for all meta v1 comma meta v2 indeed parenthesis 
  0 < 1/2 * meta ep imply meta n1 <= meta v1 imply meta n1 <= meta v2 imply 
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < 1/2 * meta ep and0
  | [ meta fy ; meta v1 ] - [ meta fy ; meta v2 ] | < 1/2 * meta ep end parenthesis imply
meta n2 <= meta m imply [ meta fx ; meta m ] !=  [ meta fy ; meta m ] end line

line ell g because pred lemma EAE mp modus ponens ell f modus ponens ell c indeed
for all meta v1 comma meta v2 indeed parenthesis 
  0 < 1/2 * meta ep imply meta n1 <= meta v1 imply meta n1 <= meta v2 imply 
  | [ meta fx ; meta v1 ] - [ meta fx ; meta v2 ] | < 1/2 * meta ep and0
  | [ meta fy ; meta v1 ] - [ meta fy ; meta v2 ] | < 1/2 * meta ep end parenthesis imply
meta n2 <= meta m imply [ meta fx ; meta m ] !=  [ meta fy ; meta m ] end line

line ell h because pred lemma exist mp modus ponens ell g modus ponens ell e indeed
  meta n2 <= meta m imply [ meta fx ; meta m ] !=  [ meta fy ; meta m ] end line

line ell i because 1rule gen modus ponens ell h indeed for all meta m indeed parenthesis
  meta n2 <= meta m imply [ meta fx ; meta m ] !=  [ meta fy ; meta m ] end parenthesis end line

because pred lemma intro exist at meta n2 modus ponens ell i indeed exist0 meta n indeed 
  for all meta m indeed parenthesis meta n <= meta m imply [ meta fx ; meta m ] !=  [ meta fy ; meta m ] 
  end parenthesis qed
end math ]"





"[  math in theory system Q lemma lemma sameFreciprocal helper says
for all terms meta m comma meta n comma meta fx indeed
for all meta m indeed parenthesis meta n <= meta m imply [ meta fx ; meta m ] != [ 0f ; meta m ] end parenthesis imply
meta n <= meta m imply
[ meta fx *f 1f/ meta fx ; meta m ] = [ 1f ; meta m ]
end lemma end math ]"

"[ math system Q proof of lemma sameFreciprocal helper reads
block any term meta m comma meta n comma meta fx end line
line ell a premise for all meta m indeed parenthesis 
  meta n <= meta m imply [ meta fx ; meta m ] != [ 0f ; meta m ] end parenthesis end line
line ell b premise meta n <= meta m end line

line ell big x because lemma a4 at meta m modus ponens ell a indeed
  meta n <= meta m imply [ meta fx ; meta m ] != [ 0f ; meta m ] end line

line ell c because 1rule mp modus ponens ell big x modus ponens ell b indeed 
  [ meta fx ; meta m ] != [ 0f ; meta m ] end line

line ell x because axiom natType indeed meta m in0 N end line
line ell d because lemma 0f modus ponens ell x indeed [ 0f ; meta m ] = 0 end line
line ell e because lemma subNeqRight modus ponens ell d modus ponens ell c indeed 
  [ meta fx ; meta m ] != 0 end line

line ell f because lemma reciprocalF nonzero modus ponens ell e indeed
  [ 1f/ meta fx ; meta m ] = 1/ [ meta fx ; meta m ] end line
line ell g because lemma eqMultiplicationLeft modus ponens ell f indeed
  [ meta fx ; meta m ] * [ 1f/ meta fx ; meta m ] = [ meta fx ; meta m ] * 1/ [ meta fx ; meta m ] end line

line ell h because lemma reciprocal modus ponens ell e indeed
  [ meta fx ; meta m ] * 1/ [ meta fx ; meta m ] = 1 end line
line ell i because lemma 1f modus ponens ell x indeed [ 1f ; meta m ] = 1 end line
line ell j because lemma eqSymmetry modus ponens ell i indeed 1 = [ 1f ; meta m ] end line
line ell k because lemma eqTransitivity modus ponens ell h modus ponens ell j indeed
  [ meta fx ; meta m ] * 1/ [ meta fx ; meta m ] = [ 1f ; meta m ] end line

line ell l because lemma timesF indeed 
 [ meta fx *f 1f/ meta fx ; meta m ] = [ meta fx ; meta m ] * [ 1f/ meta fx ; meta m ] end line

because lemma eqTransitivity4 modus ponens ell l modus ponens ell g modus ponens ell k indeed 
  [ meta fx *f 1f/ meta fx ; meta m ] = [ 1f ; meta m ] end line
line ell big a end block

any term meta m comma meta n comma meta fx end line

because 1rule deduction modus ponens ell big a indeed
for all meta m indeed parenthesis meta n <= meta m imply [ meta fx ; meta m ] != [ 0f ; meta m ] end parenthesis imply
meta n <= meta m imply
[ meta fx *f 1f/ meta fx ; meta m ] = [ 1f ; meta m ] qed
end math ]"


"[  math in theory system Q lemma lemma sameFreciprocal says
for all terms meta fx indeed
not0 meta fx sameF 0f  infer  meta fx *f 1f/ meta fx  sameF  1f
end lemma end math ]"

"[ math system Q proof of lemma sameFreciprocal reads
any term meta fx end line

line ell a premise not0 meta fx sameF 0f end line
line ell b because lemma fromNotSameF(Strong) modus ponens ell a indeed
  exist0 meta n indeed for all meta m indeed parenthesis
  meta n <= meta m imply  
  [ meta fx ; meta m ] != [ 0f ; meta m ] end parenthesis end line

line ell c because lemma sameFreciprocal helper indeed
  for all meta m indeed parenthesis meta n <= meta m imply [ meta fx ; meta m ] != [ 0f ; meta m ] 
   end parenthesis imply
  meta n <= meta m imply
  [ meta fx *f 1f/ meta fx ; meta m ] = [ 1f ; meta m ] end line

line ell d because pred lemma exist mp modus ponens ell c modus ponens ell b indeed
  meta n <= meta m imply
  [ meta fx *f 1f/ meta fx ; meta m ] = [ 1f ; meta m ] end line

line ell e because 1rule gen modus ponens ell d indeed
  for all meta m indeed parenthesis meta n <= meta m imply
  [ meta fx *f 1f/ meta fx ; meta m ] = [ 1f ; meta m ] end parenthesis end line

line ell f because pred lemma intro exist at meta n modus ponens ell d indeed
  exist0 meta n indeed for all meta m indeed parenthesis meta n <= meta m imply
  [ meta fx *f 1f/ meta fx ; meta m ] = [ 1f ; meta m ] end parenthesis end line

because lemma eventually=f to sameF modus ponens ell f indeed 
  meta fx *f 1f/ meta fx sameF 1f qed
end math ]"

-------------



"[  math in theory system Q lemma lemma reciprocal(R) says
for all terms meta fx indeed
R( meta fx ) !!== 00 infer R( meta fx ) ** R( 1f/ meta fx ) == 01
end lemma end math ]"

"[ math system Q proof of lemma reciprocal(R) reads
any term meta fx end line
line ell a premise R( meta fx ) !!== 00 end line
line ell b because lemma from!!== modus ponens ell a indeed not0 meta fx sameF 0f end line
line ell c because lemma sameFreciprocal modus ponens ell b indeed
  meta fx *f 1f/ meta fx sameF 1f end line
line ell d because 1rule to== modus ponens ell c indeed R( meta fx *f 1f/ meta fx ) == R( 1f ) end line
line ell e because lemma eqReflexivity indeed 
  R( meta fx ) ** R( 1f/ meta fx ) == R( meta fx *f 1f/ meta fx ) end line

line ell f because lemma ==Transitivity modus ponens ell e modus ponens ell d indeed
  R( meta fx ) ** R( 1f/ meta fx ) == R( 1f ) end line
because 1rule repetition modus ponens ell f indeed R( meta fx ) ** R( 1f/ meta fx ) == 01 qed ""; XXREP 01
end math ]"

----------


"[  math in theory system Q lemma lemma insertTwoMiddleTerms(Sum) says
for all terms meta x comma meta y comma meta z comma meta u indeed
meta x + meta y = parenthesis meta x - meta z end parenthesis + 
                  parenthesis meta z - meta u end parenthesis + parenthesis meta u + meta y end parenthesis
end lemma end math ]"

"[ math system Q proof of lemma insertTwoMiddleTerms(Sum) reads
any term meta x comma meta y comma meta z comma meta u end line
line ell a because lemma insertMiddleTerm(Sum) indeed
  meta x + meta y = parenthesis meta x - meta z end parenthesis + 
                    parenthesis meta z + meta y end parenthesis end line

line ell b because lemma insertMiddleTerm(Sum) indeed 
  meta z + meta y = parenthesis meta z - meta u end parenthesis + 
                    parenthesis meta u + meta y end parenthesis end line
line ell c because lemma eqAdditionLeft modus ponens ell b indeed
  parenthesis meta x - meta z end parenthesis + parenthesis meta z + meta y end parenthesis =
  parenthesis meta x - meta z end parenthesis + parenthesis parenthesis meta z - meta u end parenthesis +   
    parenthesis meta u + meta y end parenthesis end parenthesis end line

line ell d because axiom plusAssociativity indeed
  parenthesis meta x - meta z end parenthesis + parenthesis meta z - meta u end parenthesis +   
  parenthesis meta u + meta y end parenthesis =  
  parenthesis meta x - meta z end parenthesis + parenthesis parenthesis meta z - meta u end parenthesis +   
  parenthesis meta u + meta y end parenthesis end parenthesis end line
line ell e because lemma eqSymmetry modus ponens ell d indeed
  parenthesis meta x - meta z end parenthesis + parenthesis parenthesis meta z - meta u end parenthesis +   
  parenthesis meta u + meta y end parenthesis end parenthesis =
  parenthesis meta x - meta z end parenthesis + parenthesis meta z - meta u end parenthesis +   
  parenthesis meta u + meta y end parenthesis end line

because lemma eqTransitivity4 modus ponens ell a modus ponens ell c modus ponens ell e indeed 
  meta x + meta y = parenthesis meta x - meta z end parenthesis + 
  parenthesis meta z - meta u end parenthesis + parenthesis meta u + meta y end parenthesis qed
end math ]"

"[  math in theory system Q lemma pred lemma EEAnegated says
for all terms meta v1 comma meta v2 comma meta v3 comma meta a indeed
not0 exist0 meta v1 indeed exist0 meta v2 indeed for all meta v3 indeed meta a infer
for all meta v1 indeed for all meta v2 indeed exist0 meta v3 indeed not0 meta a
end lemma end math ]"

"[ math system Q proof of pred lemma EEAnegated reads
any term meta v1 comma meta v2 comma meta v3 comma meta a end line

line ell big a premise 
  not0 exist0 meta v1 indeed exist0 meta v2 indeed for all meta v3 indeed meta a end line

line ell a because pred lemma allNegated(Imply) indeed
  not0 for all meta v3 indeed meta a imply exist0 meta v3 indeed not0 meta a end line

line ell b because pred lemma addAll modus ponens ell a indeed
  for all meta v2 indeed not0 for all meta v3 indeed meta a imply 
  for all meta v2 indeed exist0 meta v3 indeed not0 meta a end line
line ell c because pred lemma existNegated(Imply) indeed
  not0 exist0 meta v2 indeed for all meta v3 indeed meta a imply
  for all meta v2 indeed not0 for all meta v3 indeed meta a end line
line ell d because prop lemma imply transitivity modus ponens ell c modus ponens ell b indeed
  not0 exist0 meta v2 indeed for all meta v3 indeed meta a imply
  for all meta v2 indeed exist0 meta v3 indeed not0 meta a end line

line ell e because pred lemma addAll modus ponens ell d indeed
  for all meta v1 indeed not0 exist0 meta v2 indeed for all meta v3 indeed meta a imply
  for all meta v1 indeed for all meta v2 indeed exist0 meta v3 indeed not0 meta a end line
line ell f because pred lemma existNegated(Imply) indeed
  not0 exist0 meta v1 indeed exist0 meta v2 indeed for all meta v3 indeed meta a imply
  for all meta v1 indeed not0 exist0 meta v2 indeed for all meta v3 indeed meta a end line
line ell g because prop lemma imply transitivity modus ponens ell f modus ponens ell e indeed  
  not0 exist0 meta v1 indeed exist0 meta v2 indeed for all meta v3 indeed meta a imply
  for all meta v1 indeed for all meta v2 indeed exist0 meta v3 indeed not0 meta a end line

because 1rule mp modus ponens ell g modus ponens ell big a indeed
  for all meta v1 indeed for all meta v2 indeed exist0 meta v3 indeed not0 meta a qed
end math ]"

"[  math in theory system Q lemma lemma fromNumericalGreater says
for all terms meta x comma meta y indeed
meta x < | meta y | infer
meta y < - meta x or0 meta x < meta y
end lemma end math ]"

"[ math system Q proof of lemma fromNumericalGreater reads

block any term meta x comma meta y end line
line ell a premise meta x < | meta y | end line
line ell b premise 0 <= meta y end line
line ell c because lemma nonnegativeNumerical modus ponens ell b indeed | meta y | = meta y end line
line ell d because lemma subLessRight modus ponens ell c modus ponens ell a indeed meta x < meta y end line
because prop lemma weaken or first modus ponens ell d indeed 
  meta y < - meta x or0 meta x < meta y end line
line ell big a end block

block any term meta x comma meta y end line
line ell b premise meta x < | meta y | end line
line ell c premise meta y <= 0 end line
line ell d because lemma nonpositiveNumerical modus ponens ell c indeed | meta y | = - meta y end line
line ell e because lemma subLessRight modus ponens ell d modus ponens ell b indeed 
  meta x < - meta y end line
line ell f because lemma lessNegated modus ponens ell e indeed - - meta y < - meta x end line
line ell g because lemma doubleMinus indeed - - meta y = meta y end line
line ell h because lemma subLessLeft modus ponens ell g modus ponens ell f indeed 
  meta y < - meta x end line
because prop lemma weaken or second modus ponens ell h indeed 
  meta y < - meta x or0 meta x < meta y end line
line ell big b end block

any term meta x comma meta y end line

line ell a because 1rule deduction modus ponens ell big a indeed
  meta x < | meta y | imply 0 <= meta y imply meta y < - meta x or0 meta x < meta y end line

line ell b because 1rule deduction modus ponens ell big b indeed
  meta x < | meta y | imply meta y <= 0 imply meta y < - meta x or0 meta x < meta y end line

line ell c premise meta x < | meta y | end line

line ell d because 1rule mp modus ponens ell a modus ponens ell c indeed
  0 <= meta y imply meta y < - meta x or0 meta x < meta y end line
line ell e because 1rule mp modus ponens ell b modus ponens ell c indeed
  meta y <= 0 imply meta y < - meta x or0 meta x < meta y end line

because lemma from leqGeq modus ponens ell d modus ponens ell e indeed 
  meta y < - meta x or0 meta x < meta y qed
end math ]"




"[  math in theory system Q lemma lemma fromNot<f(Weak) helper says
for all terms meta m comma meta n comma meta ep comma meta fx comma meta fy indeed
  not0 parenthesis 0 < meta ep and0 parenthesis meta n <= meta m imply 
  [ meta fx ; meta m ] <= [ meta fy ; meta m ] - meta ep end parenthesis end parenthesis imply
  0 < meta ep imply
  meta n <= meta m and0 [ meta fy ; meta m ] - meta ep < [ meta fx ; meta m ] 
end lemma end math ]"

"[ math system Q proof of lemma fromNot<f(Weak) helper reads
block any term meta m comma meta n comma meta ep comma meta fx comma meta fy end line
line ell a premise not0 parenthesis 0 < meta ep and0 parenthesis meta n <= meta m imply 
  [ meta fx ; meta m ] <= [ meta fy ; meta m ] - meta ep end parenthesis end parenthesis end line
line ell b premise 0 < meta ep end line
line ell c because prop lemma from negated and modus ponens ell a modus ponens ell b indeed
  not0 parenthesis meta n <= meta m imply 
  [ meta fx ; meta m ] <= [ meta fy ; meta m ] - meta ep end parenthesis end line

line ell d because prop lemma from negated imply modus ponens ell c indeed 
  meta n <= meta m and0 not0 [ meta fx ; meta m ] <= [ meta fy ; meta m ] - meta ep end line
line ell e because prop lemma first conjunct modus ponens ell d indeed meta n <= meta m end line
line ell f because prop lemma second conjunct modus ponens ell d indeed
  not0 [ meta fx ; meta m ] <= [ meta fy ; meta m ] - meta ep end line
line ell g because lemma toLess modus ponens ell f indeed
  [ meta fy ; meta m ] - meta ep < [ meta fx ; meta m ] end line
because prop lemma join conjuncts modus ponens ell e modus ponens ell g indeed
  meta n <= meta m and0 [ meta fy ; meta m ] - meta ep < [ meta fx ; meta m ] end line 
line ell big a end block

any term meta m comma meta n comma meta ep comma meta fx comma meta fy end line

because 1rule deduction modus ponens ell big a indeed 
  not0 parenthesis 0 < meta ep and0 parenthesis meta n <= meta m imply 
  [ meta fx ; meta m ] <= [ meta fy ; meta m ] - meta ep end parenthesis end parenthesis imply
  0 < meta ep imply
  meta n <= meta m and0 [ meta fy ; meta m ] - meta ep < [ meta fx ; meta m ] qed
end math ]"

"[  math in theory system Q lemma lemma fromNot<f(Weak) says
for all terms meta m comma meta n comma meta ep comma meta fx comma meta fy indeed
not0 meta fx <f meta fy infer
for all meta ep indeed for all meta n indeed exist0 meta m indeed parenthesis 0 < meta ep imply  
meta n <= meta m and0 [ meta fy ; meta m ] - meta ep < [ meta fx ; meta m ] end parenthesis
end lemma end math ]"

"[ math system Q proof of lemma fromNot<f(Weak) reads

block any term meta m comma meta n comma meta ep comma meta fx comma meta fy end line
line ell a premise exist0 meta ep indeed exist0 meta n indeed for all meta m indeed 0 < meta ep and0 
  parenthesis meta n <= meta m imply [ meta fx ; meta m ] <= [ meta fy ; meta m ] - meta ep end parenthesis
  end line
line ell x because 1rule deduction modus ponens ell a indeed ""; XXded
  exist0 object ep indeed exist0 object n indeed for all object m indeed 0 < object ep and0 
  ( object n <= object m imply [ meta fx ; object m ] <= [ meta fy ; object m ] - object ep ) end line
because 1rule repetition modus ponens ell x indeed meta fx <f meta fy end line ""; XXREP <f
line ell big a end block

any term meta m comma meta n comma meta ep comma meta fx comma meta fy end line

line ell a because 1rule deduction modus ponens ell big a indeed
  exist0 meta ep indeed exist0 meta n indeed for all meta m indeed 0 < meta ep and0 
  parenthesis meta n <= meta m imply [ meta fx ; meta m ] <= [ meta fy ; meta m ] - meta ep end parenthesis
  imply meta fx <f meta fy end line

line ell b premise not0 meta fx <f meta fy end line

line ell c because prop lemma mt modus ponens ell a modus ponens ell b indeed
  not0 exist0 meta ep indeed exist0 meta n indeed for all meta m indeed 0 < meta ep and0 
  parenthesis meta n <= meta m imply [ meta fx ; meta m ] <= [ meta fy ; meta m ] - meta ep end parenthesis
  end line

line ell d because pred lemma EEAnegated modus ponens ell c indeed 
  for all meta ep indeed for all meta n indeed exist0 meta m indeed 
  not0 parenthesis 0 < meta ep and0 parenthesis meta n <= meta m imply 
  [ meta fx ; meta m ] <= [ meta fy ; meta m ] - meta ep end parenthesis end parenthesis end line

line ell e because lemma a4 at meta ep modus ponens ell d indeed 
  for all meta n indeed exist0 meta m indeed not0 parenthesis 0 < meta ep and0 
  parenthesis meta n <= meta m imply 
  [ meta fx ; meta m ] <= [ meta fy ; meta m ] - meta ep end parenthesis end parenthesis end line

line ell f because lemma a4 at meta n modus ponens ell e indeed exist0 meta m indeed 
  not0 parenthesis 0 < meta ep and0 parenthesis meta n <= meta m imply 
  [ meta fx ; meta m ] <= [ meta fy ; meta m ] - meta ep end parenthesis end parenthesis end line

line ell g because lemma fromNot<f(Weak) helper indeed  
  not0 parenthesis 0 < meta ep and0 parenthesis meta n <= meta m imply 
  [ meta fx ; meta m ] <= [ meta fy ; meta m ] - meta ep end parenthesis end parenthesis imply
  0 < meta ep imply
   meta n <= meta m and0 [ meta fy ; meta m ] - meta ep < [ meta fx ; meta m ] end line

line ell h because pred lemma exist mp modus ponens ell g modus ponens ell f indeed
   0 < meta ep imply meta n <= meta m and0 [ meta fy ; meta m ] - meta ep < [ meta fx ; meta m ] end line

line ell i because pred lemma intro exist at meta m modus ponens ell h indeed exist0 meta m indeed   
  parenthesis 0 < meta ep imply meta n <= meta m and0 [ meta fy ; meta m ] - meta ep < [ meta fx ; meta m ] 
  end parenthesis end line

line ell j because 1rule gen modus ponens ell i indeed for all meta n indeed exist0 meta m indeed   
  parenthesis 0 < meta ep imply meta n <= meta m and0 [ meta fy ; meta m ] - meta ep < [ meta fx ; meta m ] 
  end parenthesis end line

because 1rule gen modus ponens ell j indeed
  for all meta ep indeed for all meta n indeed exist0 meta m indeed   
  parenthesis 0 < meta ep imply meta n <= meta m and0 [ meta fy ; meta m ] - meta ep < [ meta fx ; meta m ]  
  end parenthesis qed
end math ]"

(******)



---------------


"[ math var a end math ""{ STOR FYLDER BEGYNDER











"[  math in theory system Q lemma lemma leqTotality(R) says
for all terms meta fx comma meta fy indeed
R( meta fx ) <<== R( meta fy ) or0 R( meta fy ) <<== R( meta fx )
end lemma end math ]"

"[ math system Q proof of lemma leqTotality(R) reads

block any term meta fx comma meta fy end line
line ell a premise not0 R( meta fx ) <<== R( meta fy ) end line
line ell b because lemma toLess(R) modus ponens ell a indeed
  R( meta fy ) << R( meta fx ) end line

because lemma lessLeq(R) modus ponens ell b indeed R( meta fy ) <<== R( meta fx ) end line
line ell big a end block

any term meta fx comma meta fy end line
line ell a because 1rule deduction modus ponens ell big a indeed 
  not0 R( meta fx ) <<== R( meta fy ) imply R( meta fy ) <<== R( meta fx ) end line
because 1rule repetition modus ponens ell a indeed ""; XXREP or
  R( meta fx ) <<== R( meta fy ) or0 R( meta fy ) <<== R( meta fx ) qed
end math ]"

"[ math in theory system Q lemma lemma positiveToLeft(Eq) says
for all terms meta x comma meta y comma meta z indeed
meta x = meta y + meta z infer meta x - meta z = meta y
end lemma end math ]"

"[ math system Q proof of lemma positiveToLeft(Eq) reads
any term meta x comma meta y comma meta z end line
line ell a premise meta x = meta y + meta z end line
line ell b because lemma eqAddition modus ponens ell a indeed 
  meta x - meta z = meta y + meta z - meta z end line

line ell c because lemma x=x+y-y indeed
  meta y = meta y + meta z - meta z end line
line ell d because lemma eqSymmetry modus ponens ell c indeed meta y + meta z - meta z = meta y end line

because lemma eqTransitivity modus ponens ell b modus ponens ell d indeed
  meta x - meta z = meta y qed
end math ]"




--------


"[  math in theory system Q lemma lemma expZero exact says
for all terms meta x indeed
meta x ^ 0 = 1 
end lemma end math ]"

"[ math system Q proof of lemma expZero exact reads
any term meta x end line
line ell a because lemma eqReflexivity indeed 0 = 0 end line
because 1rule expZero modus ponens ell a indeed meta x ^ 0 = 1 qed
end math ]"

"[  math in theory system Q lemma lemma +1IsPositive(N) says
for all terms meta m indeed Nat( meta m ) endorse 
0 < meta m + 1
end lemma end math ]"

"[ math system Q proof of lemma +1IsPositive(N) reads
any term meta m end line

line ell a side condition Nat( meta m ) end line
line ell b because axiom nonnegative(N) modus probans ell a indeed 0 <= meta m end line

because lemma leqPlus1 modus ponens ell b indeed 0 < meta m + 1 qed
end math ]" 


"[  math in theory system Q lemma lemma sameExp base says
for all terms meta n comma meta x indeed
for all meta n indeed parenthesis 0 = meta n imply meta x ^ 0 = meta x ^ meta n end parenthesis
end lemma end math ]"

"[ math system Q proof of lemma sameExp base reads
block any term meta n comma meta x end line
line ell a premise 0 = meta n end line
line ell b because lemma expZero exact indeed meta x ^ 0 = 1 end line

line ell c because lemma eqSymmetry modus ponens ell a indeed meta n = 0 end line
line ell d because 1rule expZero modus ponens ell c indeed meta x ^ meta n = 1 end line
line ell e because lemma eqSymmetry modus ponens ell d indeed 1 = meta x ^ meta n end line

because lemma eqTransitivity modus ponens ell b modus ponens ell e indeed
  meta x ^ 0 = meta x ^ meta n end line
line ell big a end block

any term meta n comma meta x end line

line ell a because 1rule deduction modus ponens ell big a indeed
  0 = meta n imply meta x ^ 0 = meta x ^ meta n end line
because 1rule gen modus ponens ell a indeed 
  for all meta n indeed parenthesis 0 = meta n imply meta x ^ 0 = meta x ^ meta n end parenthesis qed

end math ]"

"[  math in theory system Q lemma lemma sameExp indu says
for all terms meta m comma meta n comma meta x indeed
for all meta n indeed 
  parenthesis meta m = meta n imply meta x ^ meta m = meta x ^ meta n end parenthesis imply
for all meta n indeed parenthesis 
meta m + 1 = meta n imply meta x ^ parenthesis meta m + 1 end parenthesis = meta x ^ meta n end parenthesis
end lemma end math ]"

"[ math system Q proof of lemma sameExp indu reads

block any term meta m comma meta n comma meta x end line
block any term meta m comma meta n comma meta x end line
line ell a premise for all meta n indeed 
  parenthesis meta m = meta n imply meta x ^ meta m = meta x ^ meta n end parenthesis end line
line ell b premise meta m + 1 = meta n end line

line ell c because lemma +1IsPositive(N) indeed 0 < meta m + 1 end line
line ell f because 1rule expPositive modus ponens ell c indeed
  meta x ^ parenthesis meta m + 1 end parenthesis = 
  meta x * meta x ^ parenthesis meta m + 1 - 1 end parenthesis end line

line ell g because lemma x=x+y-y indeed meta m = meta m + 1 - 1 end line
line ell h because lemma a4 at meta m + 1 - 1 modus ponens ell a indeed
  meta m = meta m + 1 - 1 imply 
  meta x ^ meta m = meta x ^ parenthesis meta m + 1 - 1 end parenthesis end line
line ell i because 1rule mp modus ponens ell h modus ponens ell g indeed
  meta x ^ meta m = meta x ^ parenthesis meta m + 1 - 1 end parenthesis end line
line ell j because lemma eqSymmetry modus ponens ell i indeed
  meta x ^ parenthesis meta m + 1 - 1 end parenthesis = meta x ^ meta m end line
line ell k because lemma eqMultiplicationLeft modus ponens ell j indeed
  meta x * meta x ^ parenthesis meta m + 1 - 1 end parenthesis = meta x * meta x ^ meta m end line

line ell l because lemma a4 at meta n - 1 modus ponens ell a indeed
  meta m = meta n - 1 imply meta x ^ meta m = meta x ^ parenthesis meta n - 1 end parenthesis end line
line ell m because lemma positiveToRight(Eq) modus ponens ell b indeed
  meta m = meta n - 1 end line
line ell o because 1rule mp modus ponens ell l modus ponens ell m indeed
  meta x ^ meta m = meta x ^ parenthesis meta n - 1 end parenthesis end line
line ell p because lemma eqMultiplicationLeft modus ponens ell o indeed
  meta x * meta x ^ meta m = meta x * meta x ^ parenthesis meta n - 1 end parenthesis end line


line ell q because lemma subLessRight modus ponens ell b modus ponens ell c indeed 0 < meta n end line
line ell r because 1rule expPositive modus ponens ell q indeed 
  meta x ^ meta n = meta x * meta x ^ parenthesis meta n - 1 end parenthesis end line
line ell s because lemma eqSymmetry modus ponens ell r indeed
  meta x * meta x ^ parenthesis meta n - 1 end parenthesis = meta x ^ meta n end line

because lemma eqTransitivity5 modus ponens ell f modus ponens ell k modus ponens ell p modus ponens ell s
  indeed meta x ^ parenthesis meta m + 1 end parenthesis = meta x ^ meta n end line
line ell big a end block

line ell a because 1rule deduction modus ponens ell big a indeed 
for all meta n indeed 
  parenthesis meta m = meta n imply meta x ^ meta m = meta x ^ meta n end parenthesis imply
meta m + 1 = meta n imply
meta x ^ parenthesis meta m + 1 end parenthesis = meta x ^ meta n end line

line ell b premise for all meta n indeed 
  parenthesis meta m = meta n imply meta x ^ meta m = meta x ^ meta n end parenthesis end line

line ell c because 1rule mp modus ponens ell a modus ponens ell b indeed
  meta m + 1 = meta n imply meta x ^ parenthesis meta m + 1 end parenthesis = meta x ^ meta n end line

because 1rule gen modus ponens ell c indeed for all meta n indeed parenthesis meta m + 1 = meta n imply
  meta x ^ parenthesis meta m + 1 end parenthesis = meta x ^ meta n end parenthesis end line
line ell big b end block

any term meta m comma meta n comma meta x end line 

because 1rule deduction modus ponens ell big b indeed  
for all meta n indeed 
  parenthesis meta m = meta n imply meta x ^ meta m = meta x ^ meta n end parenthesis imply
for all meta n indeed parenthesis meta m + 1 = meta n imply
  meta x ^ parenthesis meta m + 1 end parenthesis = meta x ^ meta n end parenthesis qed
end math ]"

"[  math in theory system Q lemma lemma sameExp says
for all terms meta m comma meta n comma meta x indeed
  meta m = meta n infer meta x ^ meta m = meta x ^ meta n
end lemma end math ]"

"[ math system Q proof of lemma sameExp reads

any term meta m comma meta n comma meta x end line

line ell big a premise meta m = meta n end line
line ell a because lemma sameExp base indeed
  for all meta n indeed parenthesis 0 = meta n imply meta x ^ 0 = meta x ^ meta n end parenthesis
  end line
line ell b because lemma sameExp indu indeed for all meta n indeed 
  parenthesis meta m = meta n imply meta x ^ meta m = meta x ^ meta n end parenthesis imply
  for all meta n indeed parenthesis  meta m + 1 = meta n imply 
  meta x ^ parenthesis meta m + 1 end parenthesis = meta x ^ meta n end parenthesis end line
line ell c because lemma induction modus ponens ell a modus ponens ell b indeed
  for all meta n indeed parenthesis meta m = meta n imply meta x ^ meta m = meta x ^ meta n end parenthesis
  end line
line ell d because lemma a4 at meta n modus ponens ell c indeed 
  meta m = meta n imply meta x ^ meta m = meta x ^ meta n end line

because 1rule mp modus ponens ell d modus ponens ell big a indeed meta x ^ meta m = meta x ^ meta n qed
end math ]"

------------

"[  math in theory system Q lemma lemma exp(+1) says
for all terms meta m comma meta x indeed
meta x ^ parenthesis meta m + 1 end parenthesis = meta x * meta x ^ meta m 
end lemma end math ]"

"[ math system Q proof of lemma exp(+1) reads
any term meta m comma meta x end line
line ell a because lemma +1IsPositive(N) indeed 0 < meta m + 1 end line
line ell b because 1rule expPositive modus ponens ell a indeed
  meta x ^ parenthesis meta m + 1 end parenthesis = 
  meta x * meta x ^ parenthesis meta m + 1 - 1 end parenthesis end line

line ell c because lemma x=x+y-y indeed meta m = meta m + 1 - 1 end line
line ell d because lemma sameExp modus ponens ell c indeed
  meta x ^ meta m = meta x ^ parenthesis meta m + 1 - 1 end parenthesis end line
line ell e because lemma eqMultiplicationLeft modus ponens ell d indeed
  meta x * meta x ^ meta m = meta x * meta x ^ parenthesis meta m + 1 - 1 end parenthesis end line
line ell f because lemma eqSymmetry modus ponens ell e indeed
   meta x * meta x ^ parenthesis meta m + 1 - 1 end parenthesis = meta x * meta x ^ meta m end line

because lemma eqTransitivity modus ponens ell b modus ponens ell f indeed 
 meta x ^ parenthesis meta m + 1 end parenthesis = meta x * meta x ^ meta m qed

end math ]"





"[  math in theory system Q lemma lemma distributionOut(Minus) says
for all terms meta x comma meta y comma meta z indeed
meta x * meta y - meta x * meta z = meta x * parenthesis meta y - meta z end parenthesis
end lemma end math ]"

"[ math system Q proof of lemma distributionOut(Minus) reads

any term meta x comma meta y comma meta z end line
line ell a because lemma times(-1)Left indeed 
 (-1) * parenthesis meta x * meta z end parenthesis = - parenthesis meta x * meta z end parenthesis end line
line ell b because lemma eqSymmetry modus ponens ell a indeed
 - parenthesis meta x * meta z end parenthesis = (-1) * parenthesis meta x * meta z end parenthesis end line

line ell c because axiom timesCommutativity indeed
  (-1) * parenthesis meta x * meta z end parenthesis = meta x * meta z * (-1) end line

line ell d because axiom timesAssociativity indeed
  meta x * meta z * (-1) = meta x * parenthesis meta z * (-1) end parenthesis end line

line ell e because lemma times(-1) indeed meta z * (-1) = - meta z end line
line ell f because lemma eqMultiplicationLeft modus ponens ell e indeed
 meta x * parenthesis meta z * (-1) end parenthesis = meta x * parenthesis - meta z end parenthesis end line

line ell g because lemma eqTransitivity5 modus ponens ell b modus ponens ell c modus ponens ell d
  modus ponens ell f indeed - parenthesis meta x * meta z end parenthesis = 
  meta x * parenthesis - meta z end parenthesis end line

line ell h because lemma eqAdditionLeft modus ponens ell g indeed
  meta x * meta y - meta x * meta z = meta x * meta y + meta x * parenthesis - meta z end parenthesis end line

line ell i because lemma distributionOut indeed 
  meta x * meta y + meta x * parenthesis - meta z end parenthesis = 
  meta x * parenthesis meta y - meta z end parenthesis end line

because lemma eqTransitivity modus ponens ell h modus ponens ell i indeed 
  meta x * meta y - meta x * meta z = meta x * parenthesis meta y - meta z end parenthesis qed
end math ]"

"[ math in theory system Q lemma lemma (1/2)(x+y)-x=(1/2)(y-x) says
for all terms meta x comma meta y indeed
1/2 * parenthesis meta x + meta y end parenthesis - meta x = 
1/2 * parenthesis meta y - meta x end parenthesis
end lemma end math ]"

"[ math system Q proof of lemma (1/2)(x+y)-x=(1/2)(y-x) reads
any term meta x comma meta y end line
line ell a because axiom distribution indeed 
  1/2 * parenthesis meta x + meta y end parenthesis = 1/2 * meta x + 1/2 * meta y end line
line ell b because lemma eqAddition modus ponens ell a indeed
  1/2 * parenthesis meta x + meta y end parenthesis - meta x = 1/2 * meta x + 1/2 * meta y - meta x end line

line ell c because axiom plusCommutativity indeed 
  1/2 * meta x + 1/2 * meta y = 1/2 * meta y + 1/2 * meta x end line
line ell d because lemma eqAddition modus ponens ell c indeed 
  1/2 * meta x + 1/2 * meta y - meta x = 1/2 * meta y + 1/2 * meta x - meta x end line

line ell big a because axiom plusAssociativity indeed
  1/2 * meta y + 1/2 * meta x - meta x = 
  1/2 * meta y + parenthesis 1/2 * meta x - meta x end parenthesis end line

line ell e because lemma (1/2)x+(1/2)x=x indeed 1/2 * meta x + 1/2 * meta x = meta x end line
line ell f because lemma positiveToRight(Eq) modus ponens ell e indeed 
  1/2 * meta x = meta x - 1/2 * meta x end line
line ell g because lemma eqNegated modus ponens ell f indeed
  - parenthesis 1/2 * meta x end parenthesis = - parenthesis meta x - 1/2 * meta x end parenthesis end line

line ell h because lemma minusNegated indeed 
  - parenthesis meta x - 1/2 * meta x end parenthesis = 1/2 * meta x - meta x end line

line ell i because lemma eqTransitivity modus ponens ell g modus ponens ell h indeed
  - parenthesis 1/2 * meta x end parenthesis = 1/2 * meta x - meta x end line
line ell j because lemma eqSymmetry modus ponens ell i indeed
  1/2 * meta x - meta x = - parenthesis 1/2 * meta x end parenthesis end line
line ell k because lemma eqAdditionLeft modus ponens ell j indeed
  1/2 * meta y + parenthesis 1/2 * meta x - meta x end parenthesis = 1/2 * meta y - 1/2 * meta x end line

line ell l because lemma distributionOut(Minus) indeed
  1/2 * meta y - 1/2 * meta x = 1/2 * parenthesis meta y - meta x end parenthesis end line

because lemma eqTransitivity6 
  modus ponens ell b modus ponens ell d modus ponens ell big a modus ponens ell k modus ponens ell l indeed 
  1/2 * parenthesis meta x + meta y end parenthesis - meta x = 
  1/2 * parenthesis meta y - meta x end parenthesis qed
end math ]"


"[  math in theory system Q lemma lemma y-(1/2)(x+y)=(1/2)(y-x) says
for all terms meta x comma meta y indeed
  meta y - 1/2 * parenthesis meta x + meta y end parenthesis = 
  1/2 * parenthesis meta y - meta x end parenthesis
end lemma end math ]"

"[ math system Q proof of lemma y-(1/2)(x+y)=(1/2)(y-x) reads
any term meta x comma meta y end line
line ell a because axiom distribution indeed 
  1/2 * parenthesis meta x + meta y end parenthesis = 1/2 * meta x + 1/2 * meta y end line
line ell b because lemma eqNegated modus ponens ell a indeed
  - parenthesis 1/2 * parenthesis meta x + meta y end parenthesis end parenthesis = 
  - parenthesis 1/2 * meta x + 1/2 * meta y end parenthesis end line
line ell c because lemma eqAdditionLeft modus ponens ell b indeed
  meta y - 1/2 * parenthesis meta x + meta y end parenthesis = 
  meta y - parenthesis 1/2 * meta x + 1/2 * meta y end parenthesis end line

line ell d because axiom plusCommutativity indeed 
  1/2 * meta x + 1/2 * meta y = 1/2 * meta y + 1/2 * meta x end line
line ell e because lemma eqNegated modus ponens ell d indeed
  - parenthesis 1/2 * meta x + 1/2 * meta y end parenthesis = 
  - parenthesis 1/2 * meta y + 1/2 * meta x end parenthesis end line

line ell f because lemma -x-y=-(x+y) indeed 
  - parenthesis 1/2 * meta y end parenthesis - 1/2 * meta x = 
  - parenthesis 1/2 * meta y + 1/2 * meta x end parenthesis end line
line ell g because lemma eqSymmetry modus ponens ell f indeed
  - parenthesis 1/2 * meta y + 1/2 * meta x end parenthesis = 
  - parenthesis 1/2 * meta y end parenthesis - 1/2 * meta x end line

line ell h because lemma eqTransitivity modus ponens ell e modus ponens ell g indeed
  - parenthesis 1/2 * meta x + 1/2 * meta y end parenthesis = 
  - parenthesis 1/2 * meta y end parenthesis - 1/2 * meta x end line
line ell i because lemma eqAdditionLeft modus ponens ell h indeed
  meta y - parenthesis 1/2 * meta x + 1/2 * meta y end parenthesis = 
  meta y + parenthesis - parenthesis 1/2 * meta y end parenthesis - 1/2 * meta x end parenthesis end line
line ell big i because axiom plusAssociativity indeed
  meta y - 1/2 * meta y - 1/2 * meta x =
  meta y + parenthesis - parenthesis 1/2 * meta y end parenthesis - 1/2 * meta x end parenthesis end line
line ell big j because lemma eqSymmetry modus ponens ell big i indeed     
  meta y + parenthesis - parenthesis 1/2 * meta y end parenthesis - 1/2 * meta x end parenthesis =
  meta y - 1/2 * meta y - 1/2 * meta x end line

line ell j because lemma (1/2)x+(1/2)x=x indeed 1/2 * meta y + 1/2 * meta y = meta y end line
line ell k because lemma positiveToRight(Eq) modus ponens ell j indeed 
  1/2 * meta y = meta y - 1/2 * meta y end line
line ell l because lemma eqSymmetry modus ponens ell k indeed meta y - 1/2 * meta y = 1/2 * meta y end line
line ell m because lemma eqAddition modus ponens ell l indeed
  meta y - 1/2 * meta y - 1/2 * meta x = 1/2 * meta y - 1/2 * meta x end line

line ell n because lemma distributionOut(Minus) indeed
  1/2 * meta y - 1/2 * meta x = 1/2 * parenthesis meta y - meta x end parenthesis end line

because lemma eqTransitivity6
  modus ponens ell c modus ponens ell i modus ponens ell big j modus ponens ell m modus ponens ell n indeed 
  meta y - 1/2 * parenthesis meta x + meta y end parenthesis =
  1/2 * parenthesis meta y - meta x end parenthesis qed
end math ]"




--------------

"[  math in theory system Q lemma lemma positiveBase base says
for all terms meta x indeed
0 < meta x ^ 0
end lemma end math ]"

"[ math system Q proof of lemma positiveBase base reads
any term meta x end line
line ell a because lemma expZero exact indeed meta x ^ 0 = 1 end line
line ell b because lemma eqSymmetry modus ponens ell a indeed 1 = meta x ^ 0 end line
line ell c because lemma 0<1 indeed 0 < 1 end line

because lemma subLessRight modus ponens ell b modus ponens ell c indeed 0 < meta x ^ 0 qed
end math ]"

"[  math in theory system Q lemma lemma positiveBase indu says
for all terms meta m comma meta x indeed
0 < meta x infer 0 < meta x ^ meta m imply 0 < meta x ^ parenthesis meta m + 1 end parenthesis
end lemma end math ]"

"[ math system Q proof of lemma positiveBase indu reads
block any term meta m comma meta x end line
line ell a premise 0 < meta x end line
line ell b premise 0 < meta x ^ meta m end line
line ell c because lemma exp(+1) indeed 
  meta x ^ parenthesis meta m + 1 end parenthesis = meta x * meta x ^ meta m end line
line ell d because lemma eqSymmetry modus ponens ell c indeed
  meta x * meta x ^ meta m =  meta x ^ parenthesis meta m + 1 end parenthesis end line

line ell e because lemma positiveFactors modus ponens ell a modus ponens ell b indeed
  0 < meta x * meta x ^ meta m end line
because lemma subLessRight modus ponens ell d modus ponens ell e indeed
  0 < meta x ^ parenthesis meta m + 1 end parenthesis end line
line ell big a end block

any term meta m comma meta x end line

line ell a because 1rule deduction modus ponens ell big a indeed
  0 < meta x imply 0 < meta x ^ meta m imply 0 < meta x ^ parenthesis meta m + 1 end parenthesis end line
line ell b premise 0 < meta x end line

because 1rule mp modus ponens ell a modus ponens ell b indeed 
 0 < meta x ^ meta m imply 0 < meta x ^ parenthesis meta m + 1 end parenthesis qed
end math ]"

"[  math in theory system Q lemma lemma positiveBase says
for all terms meta m comma meta x indeed
0 < meta x infer 0 < meta x ^ meta m 
end lemma end math ]"

"[ math system Q proof of lemma positiveBase reads
any term meta m comma meta x end line
line ell a premise 0 < meta x end line
line ell b because lemma positiveBase base indeed 0 < meta x ^ 0 end line
line ell c because lemma positiveBase indu modus ponens ell a indeed
  0 < meta x ^ meta m imply 0 < meta x ^ parenthesis meta m + 1 end parenthesis end line

because lemma induction modus ponens ell b modus ponens ell c indeed 0 < meta x ^ meta m qed
end math ]"

"[  math in theory system Q lemma lemma positiveToRight(Eq)(1 term) says
for all terms meta x comma meta y indeed
meta x = meta y infer 0 = meta y - meta x
end lemma end math ]"

"[ math system Q proof of lemma positiveToRight(Eq)(1 term) reads
any term meta x comma meta y end line
line ell a premise meta x = meta y end line
line ell b because lemma eqSymmetry modus ponens ell a indeed meta y = meta x end line
line ell c because lemma positiveToLeft(Eq)(1 term) modus ponens ell b indeed
  meta y - meta x = 0 end line
because lemma eqSymmetry modus ponens ell c indeed 0 = meta y - meta x qed
end math ]"

---------

"[  math in theory system Q lemma lemma base(1/2)Sum zero exact says
for all terms meta m indeed
base(1/2)Sum( meta m , 0 ) = 1/2 ^ meta m
end lemma end math ]"

"[ math system Q proof of lemma base(1/2)Sum zero exact reads
any term meta m end line
line ell a because lemma eqReflexivity indeed 0 = 0 end line
because 1rule base(1/2)Sum zero modus ponens ell a indeed 
  base(1/2)Sum( meta m , 0 ) = 1/2 ^ meta m qed
end math ]"

"[  math in theory system Q lemma lemma sameBase(1/2)Sum second base says
for all terms meta m comma meta n2 indeed
for all meta n2 indeed parenthesis
0 = meta n2 imply base(1/2)Sum( meta m , 0 ) = base(1/2)Sum( meta m , meta n2 ) end parenthesis
end lemma end math ]"

"[ math system Q proof of lemma sameBase(1/2)Sum second base reads
block any term meta m comma meta n2 end line
line ell a premise 0 = meta n2 end line
line ell b because lemma eqSymmetry modus ponens ell a indeed meta n2 = 0 end line
line ell c because 1rule base(1/2)Sum zero modus ponens ell b indeed 
  base(1/2)Sum( meta m , meta n2 ) = 1/2 ^ parenthesis meta m end parenthesis end line
line ell d because lemma eqSymmetry modus ponens ell c indeed
  1/2 ^ parenthesis meta m end parenthesis = base(1/2)Sum( meta m , meta n2 ) end line

line ell e because lemma base(1/2)Sum zero exact indeed
  base(1/2)Sum( meta m , 0 ) = 1/2 ^ parenthesis meta m end parenthesis end line
because lemma eqTransitivity modus ponens ell e modus ponens ell d indeed
  base(1/2)Sum( meta m , 0 ) = base(1/2)Sum( meta m , meta n2 ) end line
line ell big a end block

any term meta m comma meta n2 end line
line ell a because 1rule deduction modus ponens ell big a indeed
  0 = meta n2 imply base(1/2)Sum( meta m , 0 ) = base(1/2)Sum( meta m , meta n2 ) end line


because 1rule gen modus ponens ell a indeed 
  for all meta n2 indeed parenthesis
  0 = meta n2 imply base(1/2)Sum( meta m , 0 ) = base(1/2)Sum( meta m , meta n2 ) end parenthesis qed
end math ]"

"[  math in theory system Q lemma lemma sameBase(1/2)Sum second indu says
for all terms meta m comma meta n1 comma meta n2 indeed
for all meta n2 indeed parenthesis
meta n1 = meta n2 imply base(1/2)Sum( meta m , meta n1 ) = base(1/2)Sum( meta m , meta n2 ) end parenthesis
imply
for all meta n2 indeed parenthesis
  meta n1 + 1  = meta n2 imply base(1/2)Sum( meta m , meta n1 + 1 ) = base(1/2)Sum( meta m , meta n2 ) 
  end parenthesis
end lemma end math ]"

"[ math system Q proof of lemma sameBase(1/2)Sum second indu reads
block any term meta m comma meta n1 comma meta n2 end line
block any term meta m comma meta n1 comma meta n2 end line
line ell a premise for all meta n2 indeed parenthesis
  meta n1 = meta n2 imply base(1/2)Sum( meta m , meta n1 ) = base(1/2)Sum( meta m , meta n2 ) 
  end parenthesis end line
line ell b premise meta n1 + 1 = meta n2 end line

line ell c because lemma +1IsPositive(N) indeed 0 < meta n1 + 1 end line
line ell d because 1rule base(1/2)Sum positive modus ponens ell c indeed
  base(1/2)Sum( meta m , meta n1 + 1 ) = 
  1/2 ^ parenthesis meta m + parenthesis meta n1 + 1 end parenthesis end parenthesis + 
  base(1/2)Sum( meta m , meta n1 + 1 - 1 ) end line



line ell e because lemma x=x+y-y indeed meta n1 = meta n1 + 1 - 1 end line
line ell f because lemma a4 at meta n1 + 1 - 1 modus ponens ell a indeed
  meta n1 = meta n1 + 1 - 1 imply 
  base(1/2)Sum( meta m , meta n1 ) = base(1/2)Sum( meta m , meta n1 + 1 - 1 ) end line
line ell g because 1rule mp modus ponens ell f modus ponens ell e indeed
  base(1/2)Sum( meta m , meta n1 ) = base(1/2)Sum( meta m , meta n1 + 1 - 1 ) end line
line ell h because lemma eqSymmetry modus ponens ell g indeed
  base(1/2)Sum( meta m , meta n1 + 1 - 1 ) = base(1/2)Sum( meta m , meta n1 ) end line
line ell i because lemma eqAdditionLeft modus ponens ell h indeed
  1/2 ^ parenthesis meta m + parenthesis meta n1 + 1 end parenthesis end parenthesis + 
    base(1/2)Sum( meta m , meta n1 + 1 - 1 ) = 
  1/2 ^ parenthesis meta m + parenthesis meta n1 + 1 end parenthesis end parenthesis + 
    base(1/2)Sum( meta m , meta n1 ) end line

line ell j because lemma positiveToRight(Eq) modus ponens ell b indeed meta n1 = meta n2 - 1 end line
line ell k because lemma a4 at meta n2 - 1 modus ponens ell a indeed
  meta n1 = meta n2 - 1 imply 
  base(1/2)Sum( meta m , meta n1 ) = base(1/2)Sum( meta m , meta n2 - 1 ) end line
line ell l because 1rule mp modus ponens ell k modus ponens ell j indeed
  base(1/2)Sum( meta m , meta n1 ) = base(1/2)Sum( meta m , meta n2 - 1 ) end line
line ell m because lemma eqAdditionLeft modus ponens ell l indeed
  1/2 ^ parenthesis meta m + parenthesis meta n1 + 1 end parenthesis end parenthesis + 
    base(1/2)Sum( meta m , meta n1 ) = 
  1/2 ^ parenthesis meta m + parenthesis meta n1 + 1 end parenthesis end parenthesis + 
    base(1/2)Sum( meta m , meta n2 - 1 ) end line

line ell n because lemma eqAdditionLeft modus ponens ell b indeed
  meta m + parenthesis meta n1 + 1 end parenthesis = meta m + meta n2 end line 
line ell o because lemma sameExp modus ponens ell n indeed
 1/2 ^ parenthesis meta m + parenthesis meta n1 + 1 end parenthesis end parenthesis = 
 1/2 ^ parenthesis meta m + meta n2 end parenthesis end line
line ell p because lemma eqAddition modus ponens ell o indeed
  1/2 ^ parenthesis meta m + parenthesis meta n1 + 1 end parenthesis end parenthesis + 
    base(1/2)Sum( meta m , meta n2 - 1 ) = 
  1/2 ^ parenthesis meta m + meta n2 end parenthesis + base(1/2)Sum( meta m , meta n2 - 1 ) end line

line ell q because lemma subLessRight modus ponens ell b modus ponens ell c indeed 0 < meta n2 end line
line ell r because 1rule base(1/2)Sum positive modus ponens ell q indeed
  base(1/2)Sum( meta m , meta n2 ) = 
  1/2 ^ parenthesis meta m + meta n2 end parenthesis + base(1/2)Sum( meta m , meta n2 - 1 ) end line
line ell s because lemma eqSymmetry modus ponens ell r indeed
  1/2 ^ parenthesis meta m + meta n2 end parenthesis + base(1/2)Sum( meta m , meta n2 - 1 ) = 
  base(1/2)Sum( meta m , meta n2 ) end line


because lemma eqTransitivity6 modus ponens ell d modus ponens ell i modus ponens ell m 
  modus ponens ell p modus ponens ell s indeed  
  base(1/2)Sum( meta m , meta n1 + 1 ) = base(1/2)Sum( meta m , meta n2 ) end line
line ell big a end block

line ell a because 1rule deduction modus ponens ell big a indeed
for all meta n2 indeed parenthesis
  meta n1 = meta n2 imply base(1/2)Sum( meta m , meta n1 ) = base(1/2)Sum( meta m , meta n2 ) 
  end parenthesis imply
meta n1 + 1 = meta n2 imply
base(1/2)Sum( meta m , meta n1 + 1 ) = base(1/2)Sum( meta m , meta n2 ) end line

line ell b premise for all meta n2 indeed parenthesis meta n1 = meta n2 imply 
  base(1/2)Sum( meta m , meta n1 ) = base(1/2)Sum( meta m , meta n2 ) end parenthesis end line
line ell c because 1rule mp modus ponens ell a modus ponens ell b indeed
  meta n1 + 1 = meta n2 imply
  base(1/2)Sum( meta m , meta n1 + 1 ) = base(1/2)Sum( meta m , meta n2 ) end line
because 1rule gen modus ponens ell c indeed for all meta n2 indeed parenthesis
  meta n1 + 1 = meta n2 imply
  base(1/2)Sum( meta m , meta n1 + 1 ) = base(1/2)Sum( meta m , meta n2 ) end parenthesis end line
line ell big b end block

any term meta m comma meta n1 comma meta n2 end line

because 1rule deduction modus ponens ell big b indeed
for all meta n2 indeed parenthesis meta n1 = meta n2 imply 
  base(1/2)Sum( meta m , meta n1 ) = base(1/2)Sum( meta m , meta n2 ) end parenthesis imply
for all meta n2 indeed parenthesis
  meta n1 + 1 = meta n2 imply
  base(1/2)Sum( meta m , meta n1 + 1 ) = base(1/2)Sum( meta m , meta n2 ) end parenthesis qed
end math ]"

"[  math in theory system Q lemma lemma sameBase(1/2)Sum second says
for all terms meta m comma meta n1 comma meta n2 indeed
meta n1 = meta n2 infer base(1/2)Sum( meta m , meta n1 ) = base(1/2)Sum( meta m , meta n2 )
end lemma end math ]"

"[ math system Q proof of lemma sameBase(1/2)Sum second reads
any term meta m comma meta n1 comma meta n2 end line
line ell a premise meta n1 = meta n2 end line

line ell b because lemma sameBase(1/2)Sum second base indeed for all meta n2 indeed parenthesis
  0 = meta n2 imply base(1/2)Sum( meta m , 0 ) = base(1/2)Sum( meta m , meta n2 ) end parenthesis end line
line ell c because lemma sameBase(1/2)Sum second indu indeed for all meta n2 indeed parenthesis
meta n1 = meta n2 imply base(1/2)Sum( meta m , meta n1 ) = base(1/2)Sum( meta m , meta n2 ) end parenthesis
  imply
  for all meta n2 indeed parenthesis
    meta n1 + 1  = meta n2 imply base(1/2)Sum( meta m , meta n1 + 1 ) = base(1/2)Sum( meta m , meta n2 ) 
    end parenthesis end line
line ell d because lemma induction modus ponens ell b modus ponens ell c indeed for all meta n2 indeed   
  parenthesis meta n1 = meta n2 imply base(1/2)Sum( meta m , meta n1 ) = base(1/2)Sum( meta m , meta n2 ) 
  end parenthesis end line

line ell e because lemma a4 at meta n2 modus ponens ell d indeed
  meta n1 = meta n2 imply base(1/2)Sum( meta m , meta n1 ) = base(1/2)Sum( meta m , meta n2 ) end line

because 1rule mp modus ponens ell e modus ponens ell a indeed
  base(1/2)Sum( meta m , meta n1 ) = base(1/2)Sum( meta m , meta n2 ) qed
end math ]"

--------------


"[  math in theory system Q lemma lemma negativeToLeft(Less)(1 term) says
for all terms meta x comma meta y indeed
 0 < meta x - meta y infer meta y < meta x
end lemma end math ]"

"[ math system Q proof of lemma negativeToLeft(Less)(1 term) reads
any term meta x comma meta y end line
line ell a premise 0 < meta x - meta y end line
line ell b because lemma lessAddition modus ponens ell a indeed
  0 + meta y < meta x - meta y + meta y end line

line ell c because lemma plus0Left indeed 0 + meta y = meta y end line
line ell d because lemma subLessLeft modus ponens ell c modus ponens ell b indeed
  meta y < meta x - meta y + meta y end line

line ell e because lemma three2threeTerms indeed 
  meta x - meta y + meta y = meta x + meta y - meta y end line
line ell f because lemma x=x+y-y indeed meta x = meta x + meta y - meta y end line
line ell g because lemma eqSymmetry modus ponens ell f indeed meta x + meta y - meta y = meta x end line
line ell h because lemma eqTransitivity modus ponens ell e modus ponens ell g indeed   
  meta x - meta y + meta y = meta x end line

because lemma subLessRight modus ponens ell h modus ponens ell d indeed meta y < meta x qed
end math ]"

"[  math in theory system Q lemma lemma base(1/2)Sum(+1) says
for all terms meta m comma meta n indeed
base(1/2)Sum( meta m , meta n + 1 ) = 
1/2 ^ parenthesis meta m + meta n + 1 end parenthesis + base(1/2)Sum( meta m , meta n )
end lemma end math ]"


"[ math system Q proof of lemma base(1/2)Sum(+1) reads
any term meta m comma meta n end line
line ell a because lemma +1IsPositive(N) indeed 0 < meta n + 1 end line
line ell b because 1rule base(1/2)Sum positive modus ponens ell a indeed
  base(1/2)Sum( meta m , meta n + 1 ) = 
  1/2 ^ parenthesis meta m + parenthesis meta n + 1 end parenthesis end parenthesis + 
    base(1/2)Sum( meta m , meta n + 1 - 1 ) end line

line ell f because axiom plusAssociativity indeed 
  meta m + meta n + 1 = meta m + parenthesis meta n + 1 end parenthesis end line
line ell g because lemma sameExp modus ponens ell f indeed
  1/2 ^ parenthesis meta m + meta n + 1 end parenthesis = 
  1/2 ^ parenthesis meta m + parenthesis meta n + 1 end parenthesis end parenthesis end line
line ell h because lemma eqSymmetry modus ponens ell g indeed
  1/2 ^ parenthesis meta m + parenthesis meta n + 1 end parenthesis end parenthesis =
  1/2 ^ parenthesis meta m + meta n + 1 end parenthesis end line

line ell c because lemma x=x+y-y indeed meta n = meta n + 1 - 1 end line
line ell d because lemma sameBase(1/2)Sum second modus ponens ell c indeed
  base(1/2)Sum( meta m , meta n ) = base(1/2)Sum( meta m , meta n + 1 - 1 ) end line
line ell e because lemma eqSymmetry modus ponens ell d indeed
  base(1/2)Sum( meta m , meta n + 1 - 1 ) = base(1/2)Sum( meta m , meta n ) end line

line ell i because lemma addEquations modus ponens ell h modus ponens ell e indeed
  1/2 ^ parenthesis meta m + parenthesis meta n + 1 end parenthesis end parenthesis + 
    base(1/2)Sum( meta m , meta n + 1 - 1 ) = 
  1/2 ^ parenthesis meta m + meta n + 1 end parenthesis + base(1/2)Sum( meta m , meta n ) end line

because lemma eqTransitivity modus ponens ell b modus ponens ell i indeed  
  base(1/2)Sum( meta m , meta n + 1 ) = 
  1/2 ^ parenthesis meta m + meta n + 1 end parenthesis + base(1/2)Sum( meta m , meta n ) qed
end math ]"


"[  math in theory system Q lemma lemma base(1/2)Sum exact bound base says
for all terms meta m indeed
base(1/2)Sum( meta m + 1 , 0 ) + 1/2 ^ parenthesis meta m + 1 + 0 end parenthesis = 1/2 ^ meta m
end lemma end math ]"

"[ math system Q proof of lemma base(1/2)Sum exact bound base reads
any term meta m end line
line ell a because lemma base(1/2)Sum zero exact indeed
  base(1/2)Sum( meta m + 1 , 0 ) = 1/2 ^ parenthesis meta m + 1 end parenthesis end line
line ell b because lemma exp(+1) indeed 
  1/2 ^ parenthesis meta m + 1 end parenthesis = 1/2 * 1/2 ^ meta m end line
line ell c because lemma eqTransitivity modus ponens ell a modus ponens ell b indeed
  base(1/2)Sum( meta m + 1 , 0 ) = 1/2 * 1/2 ^ meta m end line

line ell d because axiom plus0 indeed meta m + 1 + 0 = meta m + 1 end line
line ell e because lemma sameExp modus ponens ell d indeed
  1/2 ^ parenthesis meta m + 1 + 0 end parenthesis = 1/2 ^ parenthesis meta m + 1 end parenthesis end line
line ell f because lemma eqTransitivity modus ponens ell e modus ponens ell b indeed
  1/2 ^ parenthesis meta m + 1 + 0 end parenthesis = 1/2 * 1/2 ^ meta m end line

line ell g because lemma addEquations modus ponens ell c modus ponens ell f indeed
  base(1/2)Sum( meta m + 1 , 0 ) + 1/2 ^ parenthesis meta m + 1 + 0 end parenthesis = 
  1/2 * 1/2 ^ meta m + 1/2 * 1/2 ^ meta m end line

line ell h because lemma (1/2)x+(1/2)x=x indeed 
  1/2 * 1/2 ^ meta m + 1/2 * 1/2 ^ meta m = 1/2 ^ meta m end line

because lemma eqTransitivity modus ponens ell g modus ponens ell h indeed 
  base(1/2)Sum( meta m + 1 , 0 ) + 1/2 ^ parenthesis meta m + 1 + 0 end parenthesis = 1/2 ^ meta m qed
end math ]"


"[  math in theory system Q lemma lemma base(1/2)Sum exact bound indu says
for all terms meta m comma meta n indeed 
base(1/2)Sum( meta m + 1 , meta n ) + 1/2 ^ parenthesis meta m + 1 + meta n end parenthesis = 
  1/2 ^ meta m imply
base(1/2)Sum( meta m + 1 , meta n + 1 ) + 1/2 ^ parenthesis meta m + 1 + meta n + 1 end parenthesis = 
  1/2 ^ meta m
end lemma end math ]"

"[ math system Q proof of lemma base(1/2)Sum exact bound indu reads
block any term meta m comma meta n end line
line ell a premise base(1/2)Sum( meta m + 1 , meta n ) + 1/2 ^ parenthesis meta m + 1 + meta n 
  end parenthesis = 1/2 ^ meta m end line

line ell b because lemma base(1/2)Sum(+1) indeed 
  base(1/2)Sum( meta m + 1 , meta n + 1 ) = 
  1/2 ^ parenthesis meta m + 1 + meta n + 1 end parenthesis + base(1/2)Sum( meta m + 1 , meta n ) end line

line ell c because lemma eqAddition modus ponens ell b indeed  
  base(1/2)Sum( meta m + 1 , meta n + 1 ) + 
    1/2 ^ parenthesis meta m + 1 + meta n + 1 end parenthesis = 
  1/2 ^ parenthesis meta m + 1 + meta n + 1 end parenthesis + base(1/2)Sum( meta m + 1 , meta n ) + 
    1/2 ^ parenthesis meta m + 1 + meta n + 1 end parenthesis end line

line ell d because axiom plusCommutativity indeed
  1/2 ^ parenthesis meta m + 1 + meta n + 1 end parenthesis + base(1/2)Sum( meta m + 1 , meta n ) = 
  base(1/2)Sum( meta m + 1 , meta n ) + 1/2 ^ parenthesis meta m + 1 + meta n + 1 end parenthesis end line
line ell e because lemma eqAddition modus ponens ell d indeed
  1/2 ^ parenthesis meta m + 1 + meta n + 1 end parenthesis + base(1/2)Sum( meta m + 1 , meta n ) +
    1/2 ^ parenthesis meta m + 1 + meta n + 1 end parenthesis = 
  base(1/2)Sum( meta m + 1 , meta n ) + 1/2 ^ parenthesis meta m + 1 + meta n + 1 end parenthesis +
    1/2 ^ parenthesis meta m + 1 + meta n + 1 end parenthesis end line

line ell f because lemma exp(+1) indeed
  1/2 ^ parenthesis meta m + 1 + meta n + 1 end parenthesis = 
  1/2 * 1/2 ^ parenthesis meta m + 1 + meta n end parenthesis end line
line ell g because lemma addEquations modus ponens ell f modus ponens ell f indeed
  1/2 ^ parenthesis meta m + 1 + meta n + 1 end parenthesis + 
    1/2 ^ parenthesis meta m + 1 + meta n + 1 end parenthesis = 
  1/2 * 1/2 ^ parenthesis meta m + 1 + meta n end parenthesis +
    1/2 * 1/2 ^ parenthesis meta m + 1 + meta n end parenthesis end line
line ell h because lemma (1/2)x+(1/2)x=x indeed
  1/2 * 1/2 ^ parenthesis meta m + 1 + meta n end parenthesis + 
    1/2 * 1/2 ^ parenthesis meta m + 1 + meta n end parenthesis =
  1/2 ^ parenthesis meta m + 1 + meta n end parenthesis end line
line ell i because lemma eqTransitivity modus ponens ell g modus ponens ell h indeed
  1/2 ^ parenthesis meta m + 1 + meta n + 1 end parenthesis + 
    1/2 ^ parenthesis meta m + 1 + meta n + 1 end parenthesis = 
  1/2 ^ parenthesis meta m + 1 + meta n end parenthesis end line


line ell j because lemma three2twoTerms modus ponens ell i indeed
  base(1/2)Sum( meta m + 1 , meta n ) + 1/2 ^ parenthesis meta m + 1 + meta n + 1 end parenthesis +
    1/2 ^ parenthesis meta m + 1 + meta n + 1 end parenthesis =
  base(1/2)Sum( meta m + 1 , meta n ) + 1/2 ^ parenthesis meta m + 1 + meta n end parenthesis end line

because lemma eqTransitivity5
  modus ponens ell c modus ponens ell e modus ponens ell j modus ponens ell a indeed
  base(1/2)Sum( meta m + 1 , meta n + 1 ) + 1/2 ^ parenthesis meta m + 1 + meta n + 1 end parenthesis =
  1/2 ^ meta m end line
line ell big a end block

any term meta m comma meta n end line

because 1rule deduction modus ponens ell big a indeed
base(1/2)Sum( meta m + 1 , meta n ) + 1/2 ^ parenthesis meta m + 1 + meta n end parenthesis = 
  1/2 ^ meta m imply
base(1/2)Sum( meta m + 1 , meta n + 1 ) + 1/2 ^ parenthesis meta m + 1 + meta n + 1 end parenthesis = 
  1/2 ^ meta m qed
end math ]"

"[  math in theory system Q lemma lemma base(1/2)Sum exact bound says
for all terms meta m comma meta n indeed 
base(1/2)Sum( meta m + 1 , meta n ) + 1/2 ^ parenthesis meta m + 1 + meta n end parenthesis = 1/2 ^ meta m 
end lemma end math ]"


"[ math system Q proof of lemma base(1/2)Sum exact bound reads
any term meta m comma meta n end line

line ell a because lemma base(1/2)Sum exact bound base indeed
  base(1/2)Sum( meta m + 1 , 0 ) + 1/2 ^ parenthesis meta m + 1 + 0 end parenthesis = 1/2 ^ meta m end line
line ell b because lemma base(1/2)Sum exact bound indu indeed
  base(1/2)Sum( meta m + 1 , meta n ) + 1/2 ^ parenthesis meta m + 1 + meta n end parenthesis = 
    1/2 ^ meta m imply
  base(1/2)Sum( meta m + 1 , meta n + 1 ) + 1/2 ^ parenthesis meta m + 1 + meta n + 1 end parenthesis = 
    1/2 ^ meta m end line

because lemma induction modus ponens ell a modus ponens ell b indeed  
  base(1/2)Sum( meta m + 1 , meta n ) + 1/2 ^ parenthesis meta m + 1 + meta n end parenthesis = 
  1/2 ^ meta m qed
end math ]"


"[ math in theory system Q lemma lemma base(1/2)Sum bound says
for all terms meta m comma meta n indeed
base(1/2)Sum( meta m + 1 , meta n ) < 1/2 ^ meta m
end lemma end math ]"

"[ math system Q proof of lemma base(1/2)Sum bound reads
any term meta m comma meta n end line
line ell a because lemma base(1/2)Sum exact bound indeed
  base(1/2)Sum( meta m + 1 , meta n ) + 1/2 ^ parenthesis meta m + 1 + meta n end parenthesis = 
  1/2 ^ meta m end line
line ell b because axiom plusCommutativity indeed
  1/2 ^ parenthesis meta m + 1 + meta n end parenthesis + base(1/2)Sum( meta m + 1 , meta n ) =
  base(1/2)Sum( meta m + 1 , meta n ) + 1/2 ^ parenthesis meta m + 1 + meta n end parenthesis end line
line ell c because lemma eqTransitivity modus ponens ell b modus ponens ell a indeed
  1/2 ^ parenthesis meta m + 1 + meta n end parenthesis + base(1/2)Sum( meta m + 1 , meta n ) =
  1/2 ^ meta m end line
line ell d because lemma positiveToRight(Eq) modus ponens ell c indeed
  1/2 ^ parenthesis meta m + 1 + meta n end parenthesis =
  1/2 ^ meta m - base(1/2)Sum( meta m + 1 , meta n ) end line

line ell e because lemma 0<1/2 indeed 0 < 1/2 end line
line ell f because lemma positiveBase modus ponens ell e indeed
  0 < 1/2 ^ parenthesis meta m + 1 + meta n end parenthesis end line
line ell g because lemma subLessRight modus ponens ell d modus ponens ell f indeed
  0 < 1/2 ^ meta m - base(1/2)Sum( meta m + 1 , meta n ) end line

because lemma negativeToLeft(Less)(1 term) modus ponens ell g indeed 
  base(1/2)Sum( meta m + 1 , meta n ) < 1/2 ^ meta m qed
end math ]"

---------


"[  math in theory system Q lemma lemma sameSeries(NumDiff) says
for all terms meta fx comma meta fy comma meta o comma meta p comma meta n1 comma meta n2 indeed
meta o = meta p infer meta n1 = meta n2 infer
| [ meta fx ; meta o ] - [ meta fy ; meta n1 ] | = | [ meta fx ; meta p ] - [ meta fy ; meta n2 ] |
end lemma end math ]"


"[ math system Q proof of lemma sameSeries(NumDiff) reads
any term meta fx comma meta fy comma meta o comma meta p comma meta n1 comma meta n2 end line
line ell a premise meta o = meta p end line
line ell b premise meta n1 = meta n2 end line

line ell c because lemma sameSeries modus ponens ell a indeed 
  [ meta fx ; meta o ] = [ meta fx ; meta p ] end line
line ell d because lemma sameSeries modus ponens ell b indeed 
  [ meta fy ; meta n1 ] = [ meta fy ; meta n2 ] end line
line ell e because lemma eqNegated modus ponens ell d indeed
  - [ meta fy ; meta n1 ] = - [ meta fy ; meta n2 ] end line

line ell f because lemma addEquations modus ponens ell c modus ponens ell e indeed
  [ meta fx ; meta o ] - [ meta fy ; meta n1 ] = [ meta fx ; meta p ] - [ meta fy ; meta n2 ] end line

because lemma sameNumerical modus ponens ell f indeed 
  | [ meta fx ; meta o ] - [ meta fy ; meta n1 ] | = | [ meta fx ; meta p ] - [ meta fy ; meta n2 ] | qed
end math ]"

"[  math in theory system Q lemma lemma UStelescope zero exact says
for all terms meta m indeed
UStelescope( meta m , 0 ) = | [ us ; meta m ] - [ us ; meta m + 1 ] |
end lemma end math ]"

"[ math system Q proof of lemma UStelescope zero exact reads
any term meta m end line
line ell a because lemma eqReflexivity indeed 0 = 0 end line

because 1rule UStelescope zero modus ponens ell a indeed
  UStelescope( meta m , 0 ) = | [ us ; meta m ] - [ us ; meta m + 1 ] | qed
end math ]"

"[  math in theory system Q lemma lemma sameTelescope second base says
for all terms meta m comma meta n2 indeed
for all meta n2 indeed ( 0 = meta n2 imply 
  UStelescope( meta m , 0 ) = UStelescope( meta m , meta n2 ) )
end lemma end math ]"

"[ math system Q proof of lemma sameTelescope second base reads
block any term meta m comma meta n2 end line
line ell a premise 0 = meta n2 end line
line ell b because lemma eqSymmetry modus ponens ell a indeed meta n2 = 0 end line
line ell c because 1rule UStelescope zero modus ponens ell b indeed
  UStelescope( meta m , meta n2 ) = | [ us ; meta m ] - [ us ; meta m + 1 ] | end line
line ell d because lemma eqSymmetry modus ponens ell c indeed
  | [ us ; meta m ] - [ us ; meta m + 1 ] | = UStelescope( meta m , meta n2 ) end line

line ell e because lemma UStelescope zero exact indeed
  UStelescope( meta m , 0 ) = | [ us ; meta m ] - [ us ; meta m + 1 ] | end line

because lemma eqTransitivity modus ponens ell e modus ponens ell d indeed  
  UStelescope( meta m , 0 ) = UStelescope( meta m , meta n2 ) end line
line ell big a end block

any term meta m comma meta n2 end line

line ell a because 1rule deduction modus ponens ell big a indeed
  0 = meta n2 imply UStelescope( meta m , 0 ) = UStelescope( meta m , meta n2 ) end line

because 1rule gen modus ponens ell a indeed for all meta n2 indeed 
  ( 0 = meta n2 imply UStelescope( meta m , 0 ) = UStelescope( meta m , meta n2 ) ) qed
end math ]"

"[  math in theory system Q lemma lemma sameTelescope second indu says
for all terms meta m comma meta n1 comma meta n2 indeed
for all meta n2 indeed ( meta n1 = meta n2 imply 
  UStelescope( meta m , meta n1 ) = UStelescope( meta m , meta n2 ) ) imply
for all meta n2 indeed ( meta n1 + 1 = meta n2 imply
  UStelescope( meta m , meta n1 + 1 ) = UStelescope( meta m , meta n2 ) )
end lemma end math ]"

"[ math system Q proof of lemma sameTelescope second indu reads

block any term meta m comma meta n1 comma meta n2 end line
block any term meta m comma meta n1 comma meta n2 end line
line ell a premise for all meta n2 indeed ( meta n1 = meta n2 imply 
  UStelescope( meta m , meta n1 ) = UStelescope( meta m , meta n2 ) ) end line
line ell b premise meta n1 + 1 = meta n2 end line

line ell c because lemma +1IsPositive(N) indeed 0 < meta n1 + 1 end line
line ell d because 1rule UStelescope positive modus ponens ell c indeed
UStelescope( meta m , meta n1 + 1 ) = 
  | [ us ; meta m + ( meta n1 + 1 ) ] - [ us ; meta m + ( meta n1 + 1  + 1 ) ] | + 
  UStelescope( meta m , meta n1 + 1 - 1 ) end line

line ell e because lemma x=x+y-y indeed meta n1 = meta n1 + 1 - 1 end line
line ell f because lemma a4 at meta n1 + 1 - 1 modus ponens ell a indeed
  meta n1 = meta n1 + 1 - 1 imply 
  UStelescope( meta m , meta n1 ) = UStelescope( meta m , meta n1 + 1 - 1 ) end line
line ell g because 1rule mp modus ponens ell f modus ponens ell e indeed
  UStelescope( meta m , meta n1 ) = UStelescope( meta m , meta n1 + 1 - 1 ) end line
line ell h because lemma eqSymmetry modus ponens ell g indeed
   UStelescope( meta m , meta n1 + 1 - 1 ) = UStelescope( meta m , meta n1 ) end line

line ell i because lemma positiveToRight(Eq) modus ponens ell b indeed 
  meta n1 = meta n2 - 1 end line
line ell j because lemma a4 at meta n2 - 1 modus ponens ell a indeed
  meta n1 = meta n2 - 1 imply UStelescope( meta m , meta n1 ) = UStelescope( meta m , meta n2 - 1 ) end line
line ell k because 1rule mp modus ponens ell j modus ponens ell i indeed
  UStelescope( meta m , meta n1 ) = UStelescope( meta m , meta n2 - 1 ) end line
line ell l because lemma eqTransitivity modus ponens ell h modus ponens ell k indeed
 UStelescope( meta m , meta n1 + 1 - 1 ) = UStelescope( meta m , meta n2 - 1 ) end line

line ell m because lemma eqAdditionLeft modus ponens ell b indeed
  meta m + ( meta n1 + 1 ) = meta m + meta n2 end line
line ell big m because lemma eqAddition modus ponens ell b indeed
  meta n1 + 1 + 1 = meta n2 + 1 end line
line ell n because lemma eqAdditionLeft modus ponens ell big m indeed
  meta m + ( meta n1 + 1 + 1 ) = meta m + ( meta n2 + 1 ) end line
line ell o because lemma sameSeries(NumDiff) modus ponens ell m modus ponens ell n indeed
  | [ us ; meta m + ( meta n1 + 1 ) ] - [ us ; meta m + ( meta n1 + 1 + 1 ) ] | = 
  | [ us ; meta m +  meta n2 ] - [ us ; meta m + ( meta n2 + 1 ) ] | end line

line ell p because lemma addEquations modus ponens ell o modus ponens ell l indeed
  | [ us ; meta m + ( meta n1 + 1 ) ] - [ us ; meta m + ( meta n1 + 1 + 1 ) ] | + 
    UStelescope( meta m , meta n1 + 1 - 1 ) = 
  | [ us ; meta m +  meta n2 ] - [ us ; meta m + ( meta n2 + 1 ) ] | + 
    UStelescope( meta m , meta n2 - 1 ) end line

line ell q because lemma subLessRight modus ponens ell b modus ponens ell c indeed 0 < meta n2 end line
line ell r because 1rule UStelescope positive modus ponens ell q indeed
  UStelescope( meta m , meta n2 ) = 
  | [ us ; meta m + meta n2 ] - [ us ; meta m + ( meta n2 + 1 ) ] | + 
    UStelescope( meta m , meta n2 - 1 ) end line
line ell s because lemma eqSymmetry modus ponens ell r indeed
  | [ us ; meta m + meta n2 ] - [ us ; meta m + ( meta n2 + 1 ) ] | + 
    UStelescope( meta m , meta n2 - 1 ) =
  UStelescope( meta m , meta n2 ) end line
  
because lemma eqTransitivity4 modus ponens ell d modus ponens ell p modus ponens ell s indeed
  UStelescope( meta m , meta n1 + 1 ) = UStelescope( meta m , meta n2 ) end line
line ell big a end block

line ell a because 1rule deduction modus ponens ell big a indeed
for all meta n2 indeed ( meta n1 = meta n2 imply 
  UStelescope( meta m , meta n1 ) = UStelescope( meta m , meta n2 ) ) imply
meta n1 + 1 = meta n2 imply
UStelescope( meta m , meta n1 + 1 ) = UStelescope( meta m , meta n2 ) end line

line ell b premise for all meta n2 indeed ( meta n1 = meta n2 imply 
  UStelescope( meta m , meta n1 ) = UStelescope( meta m , meta n2 ) ) end line
line ell c because 1rule mp modus ponens ell a modus ponens ell b indeed
  meta n1 + 1 = meta n2 imply UStelescope( meta m , meta n1 + 1 ) = UStelescope( meta m , meta n2 ) end line
because 1rule gen modus ponens ell c indeed for all meta n2 indeed ( meta n1 + 1 = meta n2 imply   
  UStelescope( meta m , meta n1 + 1 ) = UStelescope( meta m , meta n2 ) ) end line
line ell big b end block

any term meta m comma meta n1 comma meta n2 end line

because 1rule deduction modus ponens ell big b indeed 
for all meta n2 indeed ( meta n1 = meta n2 imply 
  UStelescope( meta m , meta n1 ) = UStelescope( meta m , meta n2 ) ) imply
for all meta n2 indeed ( meta n1 + 1 = meta n2 imply
  UStelescope( meta m , meta n1 + 1 ) = UStelescope( meta m , meta n2 ) ) qed
end math ]"

"[  math in theory system Q lemma lemma sameTelescope second says
for all terms meta m comma meta n1 comma meta n2 indeed
meta n1 = meta n2 infer UStelescope( meta m , meta n1 ) = UStelescope( meta m , meta n2 ) 
end lemma end math ]"


"[ math system Q proof of lemma sameTelescope second reads
any term meta m comma meta n1 comma meta n2 end line

line ell big a premise meta n1 = meta n2 end line

line ell a because lemma sameTelescope second base indeed
  for all meta n2 indeed ( 0 = meta n2 imply 
  UStelescope( meta m , 0 ) = UStelescope( meta m , meta n2 ) ) end line
line ell b because lemma sameTelescope second indu indeed
  for all meta n2 indeed ( meta n1 = meta n2 imply 
    UStelescope( meta m , meta n1 ) = UStelescope( meta m , meta n2 ) ) imply
  for all meta n2 indeed ( meta n1 + 1 = meta n2 imply
    UStelescope( meta m , meta n1 + 1 ) = UStelescope( meta m , meta n2 ) ) end line
line ell c because lemma induction modus ponens ell a modus ponens ell b indeed
  for all meta n2 indeed ( meta n1 = meta n2 imply 
    UStelescope( meta m , meta n1 ) = UStelescope( meta m , meta n2 ) ) end line
 
line ell d because lemma a4 at meta n2 modus ponens ell c indeed
  meta n1 = meta n2 imply UStelescope( meta m , meta n1 ) = UStelescope( meta m , meta n2 ) end line
 
because 1rule mp modus ponens ell d modus ponens ell big a indeed  
  UStelescope( meta m , meta n1 ) = UStelescope( meta m , meta n2 ) qed
end math ]"

"[  math in theory system Q lemma lemma telescopeNumerical base says
for all terms meta m indeed
| [ us ; meta m ] - [ us ; meta m + ( 0 + 1 ) ] | <= UStelescope( meta m , 0 )
end lemma end math ]"

"[ math system Q proof of lemma telescopeNumerical base reads
any term meta m end line
line ell a because lemma eqReflexivity indeed 0 = 0 end line

line ell b because 1rule UStelescope zero modus ponens ell a indeed 
  UStelescope( meta m , 0 ) = | [ us ; meta m ] - [ us ; meta m + 1 ] | end line

line ell c because lemma eqReflexivity indeed meta m = meta m end line

line ell d because lemma plus0Left indeed 0 + 1 = 1 end line
line ell e because lemma eqAdditionLeft modus ponens ell d indeed meta m + ( 0 + 1 ) = meta m + 1 end line
line ell f because lemma eqSymmetry modus ponens ell e indeed meta m + 1 = meta m + ( 0 + 1 ) end line

line ell g because lemma sameSeries(NumDiff) modus ponens ell c modus ponens ell f indeed
  | [ us ; meta m ] - [ us ; meta m + 1 ] | = | [ us ; meta m ] - [ us ; meta m + ( 0 + 1 ) ] | end line

line ell h because lemma eqTransitivity modus ponens ell b modus ponens ell g indeed
  UStelescope( meta m , 0 ) = | [ us ; meta m ] - [ us ; meta m + ( 0 + 1 ) ] | end line
line ell i because lemma eqSymmetry modus ponens ell h indeed
  | [ us ; meta m ] - [ us ; meta m + ( 0 + 1 ) ] | = UStelescope( meta m , 0 ) end line

because lemma eqLeq modus ponens ell i indeed 
  | [ us ; meta m ] - [ us ; meta m + ( 0 + 1 ) ] | <= UStelescope( meta m , 0 ) qed
end math ]"

"[ math in theory system Q lemma lemma telescopeNumerical indu says
for all terms meta m comma meta n indeed
| [ us ; meta m ] - [ us ; meta m + ( meta n + 1 ) ] | <= UStelescope( meta m , meta n ) imply
| [ us ; meta m ] - [ us ; meta m + ( meta n + 1 + 1 ) ] | <= UStelescope( meta m , meta n + 1 )
end lemma end math ]"

"[ math system Q proof of lemma telescopeNumerical indu reads
block any term meta m comma meta n end line
line ell a premise 
  | [ us ; meta m ] - [ us ; meta m + ( meta n + 1 ) ] | <= UStelescope( meta m , meta n ) end line
line ell b because lemma +1IsPositive(N) indeed 0 < meta n + 1 end line
line ell c because 1rule UStelescope positive modus ponens ell b indeed
  UStelescope( meta m , meta n + 1 ) = 
  | [ us ; meta m + ( meta n + 1 ) ] - [ us ; meta m + ( meta n + 1 + 1 ) ] | +
  UStelescope( meta m , meta n + 1 - 1 ) end line

line ell d because lemma x=x+y-y indeed meta n = meta n + 1 - 1 end line 
line ell big d because lemma eqSymmetry modus ponens ell d indeed meta n + 1 - 1 = meta n end line
line ell e because lemma sameTelescope second modus ponens ell big d indeed
  UStelescope( meta m , meta n + 1 - 1 ) = UStelescope( meta m , meta n ) end line
line ell f because lemma eqAdditionLeft modus ponens ell e indeed
  | [ us ; meta m + ( meta n + 1 ) ] - [ us ; meta m + ( meta n + 1 + 1 ) ] | + 
    UStelescope( meta m , meta n + 1 - 1 ) = 
  | [ us ; meta m + ( meta n + 1 ) ] - [ us ; meta m + ( meta n + 1 + 1 ) ] | + 
    UStelescope( meta m , meta n ) end line

line ell g because lemma eqTransitivity modus ponens ell c modus ponens ell f indeed
  UStelescope( meta m , meta n + 1 ) = 
  | [ us ; meta m + ( meta n + 1 ) ] - [ us ; meta m + ( meta n + 1 + 1 ) ] | + 
    UStelescope( meta m , meta n ) end line
line ell h because lemma eqSymmetry modus ponens ell g indeed
  | [ us ; meta m + ( meta n + 1 ) ] - [ us ; meta m + ( meta n + 1 + 1 ) ] | + 
    UStelescope( meta m , meta n ) =
  UStelescope( meta m , meta n + 1 ) end line

line ell i because lemma leqAdditionLeft modus ponens ell a indeed
  | [ us ; meta m + ( meta n + 1 ) ] - [ us ; meta m + ( meta n + 1 + 1 ) ] | + 
    | [ us ; meta m ] - [ us ; meta m + ( meta n + 1 ) ] | <=  
  | [ us ; meta m + ( meta n + 1 ) ] - [ us ; meta m + ( meta n + 1 + 1 ) ] | + 
    UStelescope( meta m , meta n ) end line

line ell j because lemma subLeqRight modus ponens ell h modus ponens ell i indeed
  | [ us ; meta m + ( meta n + 1 ) ] - [ us ; meta m + ( meta n + 1 + 1 ) ] | + 
    | [ us ; meta m ] - [ us ; meta m + ( meta n + 1 ) ] | <= UStelescope( meta m , meta n + 1 ) end line

line ell k because axiom plusCommutativity indeed
  | [ us ; meta m + ( meta n + 1 ) ] - [ us ; meta m + ( meta n + 1 + 1 ) ] | + 
    | [ us ; meta m ] - [ us ; meta m + ( meta n + 1 ) ] | =
  | [ us ; meta m ] - [ us ; meta m + ( meta n + 1 ) ] | +
    | [ us ; meta m + ( meta n + 1 ) ] - [ us ; meta m + ( meta n + 1 + 1 ) ] | end line

line ell l because lemma subLeqLeft modus ponens ell k modus ponens ell j indeed
  | [ us ; meta m ] - [ us ; meta m + ( meta n + 1 ) ] | + 
  | [ us ; meta m + ( meta n + 1 ) ] - [ us ; meta m + ( meta n + 1 + 1 ) ] | <= 
    UStelescope( meta m , meta n + 1 ) end line

line ell m because lemma insertMiddleTerm(Numerical) indeed
  | [ us ; meta m ] - [ us ; meta m + ( meta n + 1 + 1 ) ] | <=
  | [ us ; meta m ] - [ us ; meta m + ( meta n + 1 ) ] | +
    | [ us ; meta m + ( meta n + 1 ) ] - [ us ; meta m + ( meta n + 1 + 1 ) ] | end line

because lemma leqTransitivity modus ponens ell m modus ponens ell l indeed  
  | [ us ; meta m ] - [ us ; meta m + ( meta n + 1 + 1 ) ] | <= UStelescope( meta m , meta n + 1 ) end line
line ell big a end block

any term meta m comma meta n end line

because 1rule deduction modus ponens ell big a indeed
  | [ us ; meta m ] - [ us ; meta m + ( meta n + 1 ) ] | <= UStelescope( meta m , meta n ) imply
  | [ us ; meta m ] - [ us ; meta m + ( meta n + 1 + 1 ) ] | <= UStelescope( meta m , meta n + 1 ) qed
end math ]"

"[  math in theory system Q lemma lemma telescopeNumerical says
for all terms meta m comma meta n indeed
| [ us ; meta m ] - [ us ; meta m + ( meta n + 1 ) ] | <= UStelescope( meta m , meta n )
end lemma end math ]"

"[ math system Q proof of lemma telescopeNumerical reads
any term meta m comma meta n end line

line ell a because lemma telescopeNumerical base indeed
  | [ us ; meta m ] - [ us ; meta m + ( 0 + 1 ) ] | <= UStelescope( meta m , 0 ) end line
line ell b because lemma telescopeNumerical indu indeed 
  | [ us ; meta m ] - [ us ; meta m + ( meta n + 1 ) ] | <= UStelescope( meta m , meta n ) imply
  | [ us ; meta m ] - [ us ; meta m + ( meta n + 1 + 1 ) ] | <= UStelescope( meta m , meta n + 1 ) end line

because lemma induction modus ponens ell a modus ponens ell b indeed 
  | [ us ; meta m ] - [ us ; meta m + ( meta n + 1 ) ] | <= UStelescope( meta m , meta n ) qed
end math ]"



---------------(21.10.06)

"[  math in theory system Q lemma prop lemma to negated double imply says
for all terms meta a comma meta b comma meta c indeed
meta a infer meta b infer not0 meta c infer not0 ( meta a imply meta b imply meta c )
end lemma end math ]"

"[ math system Q proof of prop lemma to negated double imply reads
block any term meta a comma meta b comma meta c end line
line ell a premise meta a end line
line ell b premise meta b end line
line ell c premise not0 meta c end line
line ell d premise meta a imply meta b imply meta c end line
line ell e because prop lemma mp2 modus ponens ell d modus ponens ell a modus ponens ell b indeed 
  meta c end line
because prop lemma from contradiction modus ponens ell e modus ponens ell c indeed
  not0 ( meta a imply meta b imply meta c ) end line
line ell big a end block

any term meta a comma meta b comma meta c end line

line ell a because 1rule deduction modus ponens ell big a indeed
  meta a imply meta b imply not0 meta c imply
  ( meta a imply meta b imply meta c ) imply not0 ( meta a imply meta b imply meta c ) end line
line ell b premise meta a end line
line ell c premise meta b end line
line ell d premise not0 meta c end line

line ell e because prop lemma mp3 modus ponens ell a modus ponens ell b modus ponens ell c 
  modus ponens ell d indeed 
  ( meta a imply meta b imply meta c ) imply not0 ( meta a imply meta b imply meta c ) end line

because prop lemma imply negation modus ponens ell e indeed not0 ( meta a imply meta b imply meta c ) qed
end math ]"

"[  math in theory system Q lemma lemma ==Transitivity4 says
for all terms meta rx comma meta ry comma meta rz comma meta ru indeed
meta rx == meta ry infer meta ry == meta rz infer meta rz == meta ru infer meta rx == meta ru
end lemma end math ]"

"[ math system Q proof of lemma ==Transitivity4 reads
any term meta rx comma meta ry comma meta rz comma meta ru end line
line ell a premise meta rx == meta ry end line
line ell b premise meta ry == meta rz end line
line ell c premise meta rz == meta ru end line

line ell d because lemma ==Transitivity modus ponens ell a modus ponens ell b indeed
  meta rx == meta rz end line
because lemma ==Transitivity modus ponens ell d modus ponens ell c indeed meta rx == meta ru qed
end math ]"




---------------(21.10.06)


"[  math in theory system Q lemma lemma eqAdditionLeft(R) says
for all terms meta fx comma meta fy comma meta fz indeed
meta fx == meta fy infer meta fz ++ meta fx == meta fz ++ meta fy
end lemma end math ]"

"[ math system Q proof of lemma eqAdditionLeft(R) reads
any term meta fx comma meta fy comma meta fz end line
line ell a premise meta fx == meta fy end line
line ell b because 1rule adhoc eqAddition(R) modus ponens ell a indeed 
   meta fx ++  meta fz  ==  meta fy ++  meta fz  end line
line ell c because lemma plusCommutativity(R) indeed 
   meta fz ++  meta fx ==  meta fx ++  meta fz end line
line ell d because lemma plusCommutativity(R) indeed 
   meta fy ++  meta fz ==  meta fz ++  meta fy end line

because lemma ==Transitivity4 modus ponens ell c modus ponens ell b modus ponens ell d indeed 
   meta fz ++ meta fx == meta fz ++  meta fy qed
end math ]"


"[ math in theory system Q lemma lemma x=x+(y-y)(R) says
for all terms meta fx comma meta fy indeed
meta fx == meta fx ++ ( meta fy ++ -- meta fy ) 
end lemma end math ]"

"[ math system Q proof of lemma x=x+(y-y)(R) reads
any term meta fx comma meta fy end line
line ell a because lemma plus0(R) indeed meta fx ++ 00 == meta fx end line

line ell b because lemma ==Symmetry modus ponens ell a indeed 
  meta fx == meta fx ++ 00 end line

line ell c because lemma negative(R) indeed meta fy ++ -- meta fy == 00 end line
line ell d because lemma ==Symmetry modus ponens ell c indeed 
  00 == meta fy ++ -- meta fy end line
line ell e because lemma eqAdditionLeft(R) modus ponens ell d indeed 
  meta fx ++ 00 == meta fx ++ ( meta fy ++ -- meta fy ) end line

because lemma ==Transitivity modus ponens ell b modus ponens ell e indeed  
  meta fx == meta fx ++ ( meta fy ++ -- meta fy ) qed
end math ]"


"[ math in theory system Q lemma lemma x=x+y-y(R) says
for all terms meta fx comma meta fy indeed
meta fx == meta fx ++ meta fy ++ -- meta fy  
end lemma end math ]"

"[ math system Q proof of lemma x=x+y-y(R) reads
any term meta fx comma meta fy end line
line ell a because lemma x=x+(y-y)(R) indeed
  meta fx == meta fx ++ ( meta fy  ++ -- meta fy ) end line

line ell b because lemma plusAssociativity(R) indeed 
  meta fx ++ meta fy ++ -- meta fy == 
  meta fx ++ ( meta fy ++ -- meta fy ) end line
line ell c because lemma ==Symmetry modus ponens ell b indeed
  meta fx ++ ( meta fy ++ -- meta fy ) == 
  meta fx ++ meta fy ++ -- meta fy end line

because lemma ==Transitivity modus ponens ell a modus ponens ell c indeed 
  meta fx == meta fx ++ meta fy ++ -- meta fy qed
end math ]"



----------(22.10.06)

"[ math in theory system Q lemma lemma three2twoTerms(R) says 
for all terms meta fx comma meta fy comma meta fz comma meta fu indeed 
meta fy ++ meta fz == meta fu infer 
meta fx ++ meta fy ++ meta fz == meta fx ++ meta fu 
end lemma end math ]"

"[ math system Q proof of lemma three2twoTerms(R) reads
any term meta fx comma meta fy comma meta fz comma meta fu end line
line ell a premise meta fy ++ meta fz == meta fu end line
line ell b because lemma eqAdditionLeft(R) modus ponens ell a indeed 
  meta fx ++ ( meta fy ++ meta fz ) == meta fx ++ meta fu end line
line ell c because lemma plusAssociativity(R) indeed  
  meta fx ++ meta fy ++ meta fz == meta fx ++ ( meta fy ++ meta fz ) end line
because lemma ==Transitivity modus ponens ell c modus ponens ell b indeed  
  meta fx ++ meta fy ++ meta fz == meta fx ++ meta fu qed 
end math ]"

"[  math in theory system Q lemma lemma positiveToRight(Less)(R) says
for all terms meta fx comma meta fy comma meta fz indeed
meta fx ++ meta fy << meta fz infer meta fx << meta fz ++ -- meta fy 
end lemma end math ]"

"[ math system Q proof of lemma positiveToRight(Less)(R) reads
any term meta fx comma meta fy comma meta fz end line
line ell a premise meta fx ++ meta fy << meta fz end line
line ell b because lemma lessAddition(R) modus ponens ell a indeed 
  meta fx ++ meta fy ++ -- meta fy << meta fz ++ -- meta fy end line
line ell c because lemma x=x+y-y(R) indeed 
  meta fx == meta fx ++ meta fy ++ -- meta fy end line
line ell d because lemma ==Symmetry modus ponens ell c indeed
  meta fx ++ meta fy ++ -- meta fy == meta fx end line
because lemma subLessLeft(R) modus ponens ell d modus ponens ell b indeed 
  meta fx << meta fz ++ -- meta fy qed
end math ]"



"[  math in theory system Q lemma lemma three2threeTerms(R) says
for all terms meta fx comma meta fy comma meta fz indeed
meta fx ++ meta fy ++ meta fz == meta fx ++ meta fz ++ meta fy
end lemma end math ]"

"[ math system Q proof of lemma three2threeTerms(R) reads
any term meta fx comma meta fy comma meta fz end line
line ell a because lemma plusCommutativity(R) indeed 
  meta fy ++ meta fz == meta fz ++ meta fy end line
line ell b because lemma three2twoTerms(R) modus ponens ell a indeed
  meta fx ++ meta fy ++ meta fz == meta fx ++ ( meta fz ++ meta fy ) end line

line ell c because lemma plusAssociativity(R) indeed
  meta fx ++ meta fz ++ meta fy == meta fx ++ ( meta fz ++ meta fy ) end line 
line ell d because lemma ==Symmetry modus ponens ell c indeed
  meta fx ++ ( meta fz ++ meta fy ) == meta fx ++ meta fz ++ meta fy end line

because lemma ==Transitivity modus ponens ell b modus ponens ell d indeed  
  meta fx ++ meta fy ++ meta fz == meta fx ++ meta fz ++ meta fy qed
end math ]"


--------(22.10.06)

"[ math in theory system Q lemma lemma plus0Left(R) says
for all terms meta fx indeed
00 ++ meta fx == meta fx 
end lemma end math ]"

"[ math system Q proof of lemma plus0Left(R) reads
any term meta fx end line
line ell a because lemma plus0(R) indeed meta fx ++ 00 == meta fx end line
line ell b because lemma plusCommutativity(R) indeed 00 ++ meta fx == meta fx ++ 00 end line
because lemma ==Transitivity modus ponens ell b  modus ponens ell a indeed  
  00 ++ meta fx == meta fx qed
end math ]"




"[  math in theory system Q lemma lemma positiveToRight(Eq)(R) says
for all terms meta fx comma meta fy comma meta fz indeed
meta fx ++ meta fy == meta fz infer meta fx == meta fz ++ -- meta fy
end lemma end math ]"

"[ math system Q proof of lemma positiveToRight(Eq)(R) reads
any term meta fx comma meta fy comma meta fz end line
line ell a premise meta fx ++ meta fy == meta fz end line
line ell b because 1rule adhoc eqAddition(R) modus ponens ell a indeed 
  meta fx ++ meta fy ++ -- meta fy == meta fz ++ -- meta fy end line

line ell c because lemma x=x+y-y(R) indeed
  meta fx == meta fx ++ meta fy ++ -- meta fy end line

because lemma ==Transitivity modus ponens ell c modus ponens ell b indeed  
  meta fx == meta fz ++ -- meta fy qed
end math ]"


"[  math in theory system Q lemma lemma subtractEquations(R) says
for all terms meta fx comma meta fy comma meta fz comma meta fu indeed
meta fx ++ meta fz == meta fy ++ meta fu infer meta fz == meta fu infer 
meta fx == meta fy
end lemma end math ]"

"[ math system Q proof of lemma subtractEquations(R) reads
any term meta fx comma meta fy comma meta fz comma meta fu end line
line ell b premise meta fx ++ meta fz == meta fy ++ meta fu end line
line ell a premise meta fz == meta fu end line

line ell c because 1rule adhoc eqAddition(R) modus ponens ell b indeed
  meta fx ++ meta fz ++ -- meta fz ==
  meta fy ++ meta fu ++ -- meta fz end line

line ell d because lemma plus0Left(R) indeed 00 ++ meta fz == meta fz end line
line ell e because lemma ==Transitivity modus ponens ell d modus ponens ell a indeed
  00 ++ meta fz == meta fu end line
line ell f because lemma positiveToRight(Eq)(R) modus ponens ell e indeed 
  00 == meta fu ++ -- meta fz end line
line ell g because lemma ==Symmetry modus ponens ell f indeed meta fu ++ -- meta fz == 00 end line
line ell h because lemma eqAdditionLeft(R) modus ponens ell g indeed 
  meta fy ++ ( meta fu ++ -- meta fz ) == meta fy ++ 00 end line
line ell i because lemma plusAssociativity(R) indeed
  meta fy ++ meta fu ++ -- meta fz == 
  meta fy ++ ( meta fu ++ -- meta fz ) end line
line ell j because lemma plus0(R) indeed meta fy ++ 00 == meta fy end line
line ell k because lemma ==Transitivity4 modus ponens ell i modus ponens ell h modus ponens ell j indeed
  meta fy ++ meta fu ++ -- meta fz == meta fy end line

line ell m because lemma x=x+y-y(R) indeed 
  meta fx == meta fx ++ meta fz ++ -- meta fz end line

 because lemma ==Transitivity4 modus ponens ell m modus ponens ell c modus ponens ell k indeed
  meta fx == meta fy qed
end math ]"


"[  math in theory system Q lemma lemma neqAddition(R) says
for all terms meta fx comma meta fy comma meta fz indeed
meta fx !!== meta fy infer meta fx ++ meta fz !!== meta fy ++ meta fz
end lemma end math ]"

"[ math system Q proof of lemma neqAddition(R) reads
block any term meta fx comma meta fy comma meta fz end line
line ell big a premise meta fx !!== meta fy end line
line ell big b premise meta fx ++ meta fz == meta fy ++ meta fz end line
line ell big c because lemma ==Reflexivity indeed meta fz == meta fz end line
line ell big d because lemma subtractEquations(R) modus ponens ell big b modus ponens ell big c indeed
  meta fx == meta fy end line
because prop lemma from contradiction modus ponens ell big d modus ponens ell big a indeed
  meta fx ++ meta fz !!== meta fy ++ meta fz end line
line ell big e end block

any term meta fx comma meta fy comma meta fz end line
line ell a because 1rule deduction modus ponens ell big e indeed
  meta fx !!== meta fy imply
  meta fx ++ meta fz == meta fy ++ meta fz imply 
  meta fx ++ meta fz !!== meta fy ++ meta fz  end line
line ell b premise meta fx !!== meta fy end line
line ell c because 1rule mp modus ponens ell a modus ponens ell b indeed
  meta fx ++ meta fz == meta fy ++ meta fz imply 
  meta fx ++ meta fz !!== meta fy ++ meta fz end line
because prop lemma imply negation modus ponens ell c indeed 
  meta fx ++ meta fz !!== meta fy ++ meta fz qed
end math ]"

-----------(22.10.06)


"[  math in theory system Q lemma lemma positiveToRight(Less)(R) says
for all terms meta fx comma meta fy comma meta fz indeed
meta fx ++ meta fy << meta fz infer meta fx << meta fz ++ -- meta fy
end lemma end math ]"

"[ math system Q proof of lemma positiveToRight(Less)(R) reads
any term meta fx comma meta fy comma meta fz end line
line ell a premise meta fx ++ meta fy << meta fz end line
line ell b because lemma lessAddition(R) modus ponens ell a indeed 
  meta fx ++ meta fy ++ -- meta fy << meta fz ++ -- meta fy end line
line ell c because lemma x=x+y-y(R) indeed meta fx == meta fx ++ meta fy ++ -- meta fy end line
line ell d because lemma ==Symmetry modus ponens ell c indeed
  meta fx ++ meta fy ++ -- meta fy == meta fx end line
because lemma subLessLeft(R) modus ponens ell d modus ponens ell b indeed 
  meta fx << meta fz ++ --  meta fy qed
end math ]"

"[  math in theory system Q lemma lemma positiveToRight(Less)(1 term)(R) says
for all terms meta fx comma meta fy indeed
meta fx << meta fy infer 00 << meta fy ++ -- meta fx 
end lemma end math ]"

"[ math system Q proof of lemma positiveToRight(Less)(1 term)(R) reads
any term meta fx comma meta fy end line
line ell a premise meta fx << meta fy end line
line ell b because lemma plus0Left(R) indeed 00 ++ meta fx == meta fx end line
line ell c because lemma ==Symmetry modus ponens ell b indeed meta fx == 00 ++ meta fx end line

line ell d because lemma subLessLeft(R) modus ponens ell c modus ponens ell a indeed
  00 ++ meta fx << meta fy end line

because lemma positiveToRight(Less)(R) modus ponens ell d indeed 00 << meta fy ++ -- meta fx qed
end math ]"

-----------(22.10.06)



"[  math in theory system Q lemma lemma to!!== says
for all terms meta m comma meta n comma meta ep comma meta fx comma meta fy indeed 
""; XX ekstra variable
not0 meta fx sameF meta fy infer R( meta fx ) !!== R( meta fy )
end lemma end math ]"

"[ math system Q proof of lemma to!!== reads

block any term meta m comma meta n comma meta ep comma meta fx comma meta fy end line
line ell a premise R( meta fx ) == R( meta fy ) end line
because 1rule from== modus ponens ell a indeed meta fx sameF meta fy end line
line ell big a end block

any term meta m comma meta n comma meta ep comma meta fx comma meta fy end line

line ell a because 1rule deduction modus ponens ell big a indeed
  R( meta fx ) == R( meta fy ) imply meta fx sameF meta fy end line

line ell b premise not0 meta fx sameF meta fy end line

because prop lemma mt modus ponens ell a modus ponens ell b indeed R( meta fx ) !!== R( meta fy ) qed
end math ]"



"[  math in theory system Q lemma lemma switchTerms(x<=y-z) says
for all terms meta x comma meta y comma meta z indeed
meta x <= meta y - meta z infer meta z <= meta y - meta x 
end lemma end math ]"

"[ math system Q proof of lemma switchTerms(x<=y-z) reads
any term meta x comma meta y comma meta z end line
line ell a premise meta x <= meta y - meta z end line
line ell b because lemma negativeToLeft(Leq) modus ponens ell a indeed meta x + meta z <= meta y end line
line ell c because axiom plusCommutativity indeed meta x + meta z = meta z + meta x end line
line ell d because lemma subLeqLeft modus ponens ell c modus ponens ell b indeed
  meta z + meta x <= meta y end line

because lemma positiveToRight(Leq) modus ponens ell d indeed meta z <= meta y - meta x qed
end math ]"


"[  math in theory system Q lemma pred lemma (A)to(~E~)(Imply) says
for all terms meta v1 comma meta a indeed
for all meta v1 indeed meta a imply
not0 exist0 meta v1 indeed not0 meta a 
end lemma end math ]"

"[ math system Q proof of pred lemma (A)to(~E~)(Imply) reads
block any term meta v1 comma meta a end line
line ell a premise for all meta v1 indeed meta a end line
line ell b because lemma a4 at meta v1 modus ponens ell a indeed meta a end line
line ell c because prop lemma add double neg modus ponens ell b indeed not0 not0 meta a end line
line ell d because 1rule gen modus ponens ell c indeed for all meta v1 indeed not0 not0 meta a end line
line ell e because prop lemma add double neg modus ponens ell d indeed 
  not0 not0 for all meta v1 indeed not0 not0 meta a end line
because 1rule repetition modus ponens ell e indeed 
  not0 exist0 meta v1 indeed not0 meta a end line ""; XXREP exist
line ell big a end block

any term meta v1 comma meta a end line

because 1rule deduction modus ponens ell big a indeed for all meta v1 indeed meta a imply not0 exist0 meta v1 indeed not0 meta a qed
end math ]"

"[  math in theory system Q lemma pred lemma (E)to(~A~)(Imply) says
for all terms meta v1 comma meta a indeed
exist0 meta v1 indeed meta a imply
not0 for all meta v1 indeed not0 meta a 
end lemma end math ]"

"[ math system Q proof of pred lemma (E)to(~A~)(Imply) reads
any term meta v1 comma meta a end line
line ell a because prop lemma auto imply indeed 
  not0 for all meta v1 indeed not0 meta a imply not0 for all meta v1 indeed not0 meta a end line
because 1rule repetition modus ponens ell a indeed ""; XXREP exist
  exist0 meta v1 indeed meta a imply not0 for all meta v1 indeed not0 meta a qed
end math ]"

"[  math in theory system Q lemma pred lemma (E~)to(~A)(Imply) says
for all terms meta v1 comma meta a indeed
exist0 meta v1 indeed not0 meta a imply 
not0 for all meta v1 indeed meta a 
end lemma end math ]"

"[ math system Q proof of pred lemma (E~)to(~A)(Imply) reads
block any term meta v1 comma meta a end line
line ell a premise exist0 meta v1 indeed not0 meta a end line
line ell b because prop lemma add double neg modus ponens ell a indeed 
  not0 not0 exist0 meta v1 indeed not0 meta a end line

line ell c because pred lemma (A)to(~E~)(Imply) indeed
  for all meta v1 indeed meta a imply not0 exist0 meta v1 indeed not0 meta a end line
because prop lemma mt modus ponens ell c modus ponens ell b indeed
  not0 for all meta v1 indeed meta a end line
line ell big a end block

any term meta v1 comma meta a end line

because 1rule deduction modus ponens ell big a indeed 
  exist0 meta v1 indeed not0 meta a imply not0 for all meta v1 indeed meta a qed
end math ]"



"[  math in theory system Q lemma pred lemma addNegatedAll says
for all terms meta v1 comma meta a comma meta b indeed
meta b imply meta a infer
not0 for all meta v1 indeed meta a imply not0 for all meta v1 indeed meta b
end lemma end math ]"

"[ math system Q proof of pred lemma addNegatedAll reads
any term meta v1 comma meta a comma meta b end line
line ell a premise meta b imply meta a end line
line ell b because pred lemma addAll modus ponens ell a indeed 
  for all meta v1 indeed meta b imply for all meta v1 indeed meta a end line
because prop lemma contrapositive modus ponens ell b indeed 
  not0 for all meta v1 indeed meta a imply not0 for all meta v1 indeed meta b qed
end math ]"



"[  math in theory system Q lemma pred lemma toNegatedAEA says
for all terms meta v1 comma meta v2 comma meta v3 comma meta a indeed
exist0 meta v1 indeed for all meta v2 indeed exist0 meta v3 indeed not0 meta a infer
not0 for all meta v1 indeed exist0 meta v2 indeed for all meta v3 indeed meta a 
end lemma end math ]"

"[ math system Q proof of pred lemma toNegatedAEA reads
any term meta v1 comma meta v2 comma meta v3 comma meta a end line

line ell big a premise 
  exist0 meta v1 indeed for all meta v2 indeed exist0 meta v3 indeed not0 meta a end line

line ell a because pred lemma (E~)to(~A)(Imply) indeed 
  exist0 meta v3 indeed not0 meta a imply
  not0 for all meta v3 indeed meta a end line

line ell b because pred lemma addNegatedAll modus ponens ell a indeed
  not0 for all meta v2 indeed not0 for all meta v3 indeed meta a imply
  not0 for all meta v2 indeed exist0 meta v3 indeed not0 meta a end line
line ell c because pred lemma (E)to(~A~)(Imply) indeed 
  exist0 meta v2 indeed for all meta v3 indeed meta a imply
  not0 for all meta v2 indeed not0 for all meta v3 indeed meta a end line
line ell d because prop lemma imply transitivity modus ponens ell c modus ponens ell b indeed
  exist0 meta v2 indeed for all meta v3 indeed meta a imply
  not0 for all meta v2 indeed exist0 meta v3 indeed not0 meta a end line

line ell e because pred lemma addNegatedAll modus ponens ell d indeed
  not0 for all meta v1 indeed not0 for all meta v2 indeed exist0 meta v3 indeed not0 meta a imply
  not0 for all meta v1 indeed exist0 meta v2 indeed for all meta v3 indeed meta a end line
line ell f because pred lemma (E)to(~A~)(Imply) indeed
  exist0 meta v1 indeed            for all meta v2 indeed exist0 meta v3 indeed not0 meta a imply
  not0 for all meta v1 indeed not0 for all meta v2 indeed exist0 meta v3 indeed not0 meta a end line
line ell g because prop lemma imply transitivity modus ponens ell f modus ponens ell e indeed
  exist0 meta v1 indeed  for all meta v2 indeed exist0 meta v3 indeed not0 meta a imply
  not0 for all meta v1 indeed exist0 meta v2 indeed for all meta v3 indeed meta a end line

because 1rule mp modus ponens ell g modus ponens ell big a indeed
  not0 for all meta v1 indeed exist0 meta v2 indeed for all meta v3 indeed meta a qed
end math ]"




"[  math in theory system Q lemma lemma lessNeq(F) helper says
for all terms meta m comma meta n comma meta ep comma meta fx comma meta fy indeed
for all meta m indeed 0 < meta ep and0 ( meta n <= meta m imply 
  [ meta fx ; meta m ] <= [ meta fy ; meta m ] - meta ep ) imply
not0 ( 0 < meta ep imply meta n <= meta n imply 
       | [ meta fx ; meta n ] - [ meta fy ; meta n ] | < meta ep ) end lemma end math ]"

"[ math system Q proof of lemma lessNeq(F) helper reads

block any term meta m comma meta n comma meta ep comma meta fx comma meta fy end line
line ell a premise for all meta m indeed 0 < meta ep and0 ( meta n <= meta m imply 
    [ meta fx ; meta m ] <= [ meta fy ; meta m ] - meta ep ) end line
line ell b because lemma a4 at meta n modus ponens ell a indeed
  0 < meta ep and0 ( meta n <= meta n imply 
  [ meta fx ; meta n ] <= [ meta fy ; meta n ] - meta ep ) end line

line ell c because prop lemma first conjunct modus ponens ell b indeed 0 < meta ep end line

line ell d because prop lemma second conjunct modus ponens ell b indeed
  meta n <= meta n imply [ meta fx ; meta n ] <= [ meta fy ; meta n ] - meta ep end line
line ell e because axiom leqReflexivity indeed meta n <= meta n end line
line ell f because 1rule mp modus ponens ell d modus ponens ell e indeed
  [ meta fx ; meta n ] <= [ meta fy ; meta n ] - meta ep end line
line ell g because lemma switchTerms(x<=y-z) modus ponens ell f indeed
  meta ep <= [ meta fy ; meta n ] - [ meta fx ; meta n ] end line

line ell h because lemma x<=|x| indeed 
  [ meta fy ; meta n ] - [ meta fx ; meta n ] <= | [ meta fy ; meta n ] - [ meta fx ; meta n ] | end line
line ell i because lemma leqTransitivity modus ponens ell g modus ponens ell h indeed
  meta ep <= | [ meta fy ; meta n ] - [ meta fx ; meta n ] | end line

line ell j because lemma numericalDifference indeed 
  | [ meta fy ; meta n ] - [ meta fx ; meta n ] | = | [ meta fx ; meta n ] - [ meta fy ; meta n ] | end line
line ell k because lemma subLeqRight modus ponens ell j modus ponens ell i indeed
  meta ep <= | [ meta fx ; meta n ] - [ meta fy ; meta n ] | end line

line ell l because lemma toNotLess modus ponens ell k indeed
  not0 | [ meta fx ; meta n ] - [ meta fy ; meta n ] | < meta ep end line

because prop lemma to negated double imply modus ponens ell c modus ponens ell e modus ponens ell l indeed 
  not0 ( 0 < meta ep imply meta n <= meta n imply 
         | [ meta fx ; meta n ] - [ meta fy ; meta n ] | < meta ep ) end line
line ell big a end block

any term meta m comma meta n comma meta ep comma meta fx comma meta fy end line

because 1rule deduction modus ponens ell big a indeed
for all meta m indeed 0 < meta ep and0 ( meta n <= meta m imply 
  [ meta fx ; meta m ] <= [ meta fy ; meta m ] - meta ep ) imply
not0 ( 0 < meta ep imply meta n <= meta n imply 
       | [ meta fx ; meta n ] - [ meta fy ; meta n ] | < meta ep ) qed
end math ]"


"[  math in theory system Q lemma lemma lessNeq(F) says ""; XX ekstra variable
for all terms meta m comma meta n comma meta ep comma meta fx comma meta fy indeed
meta fx <f meta fy infer not0 meta fx sameF meta fy
end lemma end math ]"

"[ math system Q proof of lemma lessNeq(F) reads
any term meta m comma meta n comma meta ep comma meta fx comma meta fy end line
line ell a premise meta fx <f meta fy end line
line ell x because 1rule repetition modus ponens ell a indeed ""; XXREP <f
  exist0 object ep indeed exist0 object n indeed for all object m indeed 
  0 < object ep and0 ( object n <= object m imply 
    [ meta fx ; object m ] <= [ meta fy ; object m ] - object ep ) end line
line ell b because 1rule deduction modus ponens ell x indeed ""; XXded
  exist0 meta ep indeed exist0 meta n indeed for all meta m indeed 
  0 < meta ep and0 ( meta n <= meta m imply 
    [ meta fx ; meta m ] <= [ meta fy ; meta m ] - meta ep ) end line

line ell c because lemma lessNeq(F) helper indeed
  for all meta m indeed 0 < meta ep and0 ( meta n <= meta m imply 
  [ meta fx ; meta m ] <= [ meta fy ; meta m ] - meta ep ) imply
  not0 ( 0 < meta ep imply meta n <= meta n imply 
         | [ meta fx ; meta n ] - [ meta fy ; meta n ] | < meta ep ) end line

line ell d because pred lemma 2exist mp modus ponens ell c modus ponens ell b indeed
  not0 ( 0 < meta ep imply meta n <= meta n imply 
         | [ meta fx ; meta n ] - [ meta fy ; meta n ] | < meta ep ) end line

line ell e because pred lemma intro exist at meta n modus ponens ell d indeed
  exist0 meta m indeed not0 ( 0 < meta ep imply meta n <= meta m imply 
  | [ meta fx ; meta m ] - [ meta fy ; meta m ] | < meta ep ) end line
line ell f because 1rule gen modus ponens ell e indeed
  for all meta n indeed  exist0 meta m indeed not0 ( 0 < meta ep imply meta n <= meta m imply 
  | [ meta fx ; meta m ] - [ meta fy ; meta m ] | < meta ep ) end line
line ell g because pred lemma intro exist at meta ep modus ponens ell f indeed
  exist0 meta ep indeed for all meta n indeed  exist0 meta m indeed 
  not0 ( 0 < meta ep imply meta n <= meta m imply 
  | [ meta fx ; meta m ] - [ meta fy ; meta m ] | < meta ep ) end line

line ell h because pred lemma toNegatedAEA modus ponens ell g indeed
  not0 for all meta ep indeed exist0 meta n indeed for all meta m indeed 
  ( 0 < meta ep imply meta n <= meta m imply 
  | [ meta fx ; meta m ] - [ meta fy ; meta m ] | < meta ep ) end line

line ell i because 1rule deduction modus ponens ell h indeed ""; XXded
  not0 for all object ep indeed exist0 object n indeed for all object m indeed 
  ( 0 < object ep imply object n <= object m imply 
  | [ meta fx ; object m ] - [ meta fy ; object m ] | < object ep ) end line

because 1rule repetition modus ponens ell i indeed not0 meta fx sameF meta fy qed ""; XXREP sameF
end math ]"

"[  math in theory system Q lemma lemma lessNeq(R) says
for all terms meta fx comma meta fy indeed
R( meta fx ) << R( meta fy ) infer R( meta fx ) !!== R( meta fy )
end lemma end math ]"

"[ math system Q proof of lemma lessNeq(R) reads
any term meta fx comma meta fy end line
line ell a premise R( meta fx ) << R( meta fy ) end line
line ell b because 1rule repetition modus ponens ell a indeed meta fx <f meta fy end line ""; XXREP <<
line ell c because lemma lessNeq(F) modus ponens ell b indeed not0 meta fx sameF meta fy end line

because lemma to!!== modus ponens ell c indeed R( meta fx ) !!== R( meta fy ) qed
end math ]"





"[  math in theory system Q lemma lemma positiveToRight(Less)(1 term) says
for all terms meta x comma meta y indeed
meta x < meta y infer 0 < meta y - meta x 
end lemma end math ]"

"[ math system Q proof of lemma positiveToRight(Less)(1 term) reads
any term meta x comma meta y end line
line ell a premise meta x < meta y end line
line ell b because lemma plus0Left indeed 0 + meta x = meta x end line
line ell c because lemma eqSymmetry modus ponens ell b indeed meta x = 0 + meta x end line
line ell d because lemma subLessLeft modus ponens ell c modus ponens ell a indeed
  0 + meta x < meta y end line

because lemma positiveToRight(Less) modus ponens ell d indeed 0 < meta y - meta x qed
end math ]"

"[  math in theory system Q lemma pred lemma (A~)to(~E) says
for all terms meta v1 comma meta a indeed
for all meta v1 indeed not0 meta a infer 
not0 exist0 meta v1 indeed meta a
end lemma end math ]"

"[ math system Q proof of pred lemma (A~)to(~E) reads
any term meta v1 comma meta a end line
line ell a premise for all meta v1 indeed not0 meta a end line
line ell b because prop lemma add double neg modus ponens ell a indeed
  not0 not0 for all meta v1 indeed not0 meta a end line

because 1rule repetition modus ponens ell b indeed not0 exist0 meta v1 indeed meta a qed ""; XXREP exist
end math ]"

"[  math in theory system Q lemma lemma toLeq(Advanced)(R) says
for all terms meta fep comma meta fx comma meta fy indeed
00 << meta fep imply meta fx ++ meta fep !!== meta fy infer
meta fy <<== meta fx
end lemma end math ]"

"[ math system Q proof of lemma toLeq(Advanced)(R) reads
block any term meta fep comma meta fx comma meta fy end line
line ell a premise meta fx << meta fy end line
line ell b because lemma positiveToRight(Less)(1 term)(R) modus ponens ell a indeed
  00 << meta fy ++ -- meta fx end line

line ell c because lemma plusCommutativity(R) indeed 
  meta fx ++ ( meta fy ++ -- meta fx ) == meta fy ++ -- meta fx ++ meta fx end line

line ell d because lemma three2threeTerms(R) indeed
  meta fy ++ -- meta fx ++ meta fx == meta fy ++ meta fx ++ -- meta fx end line

line ell e because lemma x=x+y-y(R) indeed meta fy == meta fy ++ meta fx ++ -- meta fx end line
line ell f because lemma ==Symmetry modus ponens ell e indeed 
  meta fy ++ meta fx ++ -- meta fx == meta fy end line

line ell g because lemma ==Transitivity4 modus ponens ell c modus ponens ell d modus ponens ell f indeed
  meta fx ++ ( meta fy ++ -- meta fx ) == meta fy end line

line ell h because prop lemma join conjuncts modus ponens ell b modus ponens ell g indeed
  00 << meta fy ++ -- meta fx and0 meta fx ++ ( meta fy ++ -- meta fx ) == meta fy end line

because pred lemma intro exist at meta fy ++ -- meta fx modus ponens ell h indeed
  exist0 meta fep indeed 00 << meta fep and0 meta fx ++ meta fep == meta fy end line
line ell big a end block

any term meta fep comma meta fx comma meta fy end line

line ell a because 1rule deduction modus ponens ell big a indeed meta fx << meta fy imply 
  exist0 meta fep indeed 00 << meta fep and0 meta fx ++ meta fep == meta fy end line

line ell b premise 00 << meta fep imply meta fx ++ meta fep !!== meta fy end line
line ell c because prop lemma to negated and modus ponens ell b indeed
  not0 ( 00 << meta fep and0 meta fx ++ meta fep == meta fy ) end line
line ell d because 1rule gen modus ponens ell c indeed
  for all meta fep indeed not0 ( 00 << meta fep and0 meta fx ++ meta fep == meta fy ) end line
line ell e because pred lemma (A~)to(~E) modus ponens ell d indeed
  not0 exist0 meta fep indeed 00 << meta fep and0 meta fx ++ meta fep == meta fy end line

line ell f because prop lemma mt modus ponens ell a modus ponens ell e indeed
  not0 meta fx << meta fy end line
because lemma fromNotLess(R) modus ponens ell f indeed meta fy <<== meta fx qed
end math ]"

--------(23.10.06)


"[  math in theory system Q lemma lemma leqNeqLess(R) says
for all terms meta fx comma meta fy indeed
meta fx <<== meta fy infer meta fx !!== meta fy infer meta fx << meta fy
end lemma end math ]"

"[ math system Q proof of lemma leqNeqLess(R) reads
block any term meta fx comma meta fy end line
line ell a premise meta fx <<== meta fy end line
line ell b premise meta fx !!== meta fy end line
line ell c premise meta fy <<== meta fx end line

line ell d because lemma leqAntisymmetry(R) modus ponens ell a modus ponens ell c indeed
  meta fx == meta fy end line
because prop lemma from contradiction modus ponens ell d modus ponens ell b indeed
  not0 meta fy <<== meta fx end line
line ell big a end block

any term meta fx comma meta fy end line


line ell a because 1rule deduction modus ponens ell big a indeed
  meta fx <<== meta fy imply meta fx !!== meta fy imply 
  meta fy <<== meta fx imply not0 meta fy <<== meta fx end line

line ell b premise meta fx <<== meta fy end line
line ell c premise meta fx !!== meta fy end line

line ell d because prop lemma mp2 modus ponens ell a modus ponens ell b modus ponens ell c indeed
  meta fy <<== meta fx imply not0 meta fy <<== meta fx end line

line ell e because prop lemma imply negation modus ponens ell d indeed not0 meta fy <<== meta fx end line

because lemma toLess(R) modus ponens ell e indeed meta fx << meta fy qed 
end math ]"

"[  math in theory system Q lemma lemma subLeqLeft(R) says
for all terms meta fx comma meta fy comma meta fz indeed
meta fx == meta fy infer meta fx <<== meta fz infer meta fy <<== meta fz
end lemma end math ]"

"[ math system Q proof of lemma subLeqLeft(R) reads
any term meta fx comma meta fy comma meta fz end line
line ell a premise meta fx == meta fy end line 
line ell b premise meta fx <<== meta fz end line
line ell c because lemma ==Symmetry modus ponens ell a indeed meta fy == meta fx end line
line ell d because lemma eqLeq(R) modus ponens ell c indeed meta fy <<== meta fx end line
because lemma leqTransitivity(R) modus ponens ell d modus ponens ell b indeed 
  meta fy <<== meta fz qed
end math ]"

"[  math in theory system Q lemma lemma leqLessTransitivity(R) says
for all terms meta fx comma meta fy comma meta fz indeed
meta fx <<== meta fy infer meta fy << meta fz infer meta fx << meta fz 
end lemma end math ]"

"[ math system Q proof of lemma leqLessTransitivity(R) reads
block any term meta fx comma meta fy comma meta fz end line
line ell a premise meta fx <<== meta fy end line
line ell b premise meta fy <<   meta fz end line
line ell big a premise meta fx == meta fz end line
line ell c because lemma lessLeq(R) modus ponens ell b indeed meta fy <<== meta fz end line
line ell d because lemma lessNeq(R) modus ponens ell b indeed meta fy !!== meta fz end line
line ell f because lemma subLeqLeft(R) modus ponens ell big a modus ponens ell a indeed 
  meta fz <<== meta fy end line
line ell g because lemma leqAntisymmetry(R) modus ponens ell c modus ponens ell f indeed 
  meta fy == meta fz end line
because prop lemma from contradiction modus ponens ell g modus ponens ell d indeed
  meta fx !!== meta fz end line
line ell h end block

any term meta fx comma meta fy comma meta fz end line
line ell i because 1rule deduction modus ponens ell h indeed
  meta fx <<== meta fy imply meta fy << meta fz imply meta fx == meta fz imply meta fx !!== meta fz end line
line ell j premise meta fx <<== meta fy end line
line ell k premise meta fy << meta fz end line
line ell l because prop lemma mp2 modus ponens ell i modus ponens ell j modus ponens ell k indeed
  meta fx == meta fz imply meta fx !!== meta fz end line
line ell m because prop lemma imply negation modus ponens ell l indeed
  meta fx !!== meta fz end line
line ell o because lemma lessLeq(R) modus ponens ell k indeed meta fy <<== meta fz end line
line ell q because lemma leqTransitivity(R) modus ponens ell j modus ponens ell o indeed 
  meta fx <<== meta fz end line
because lemma leqNeqLess(R) modus ponens ell q modus ponens ell m indeed meta fx << meta fz qed
end math ]"

-----------(23.10.06)

"[  math in theory system Q lemma lemma negativeToLeft(Eq)(R) says
for all terms meta fx comma meta fy comma meta fz indeed
meta fx == meta fy ++ -- meta fz infer meta fx ++ meta fz == meta fy 
end lemma end math ]"

"[ math system Q proof of lemma negativeToLeft(Eq)(R) reads
any term meta fx comma meta fy comma meta fz end line
line ell a premise meta fx == meta fy ++ -- meta fz end line
line ell b because 1rule adhoc eqAddition(R) modus ponens ell a indeed 
  meta fx ++ meta fz == meta fy ++ -- meta fz ++ meta fz end line
line ell c because lemma three2threeTerms(R) indeed
  meta fy ++ -- meta fz ++ meta fz == meta fy ++ meta fz ++ -- meta fz end line
line ell d because lemma x=x+y-y(R) indeed
  meta fy == meta fy ++ meta fz ++ -- meta fz end line
line ell e because lemma ==Symmetry modus ponens ell d indeed
  meta fy ++ meta fz ++ -- meta fz == meta fy end line

because lemma ==Transitivity4 modus ponens ell b modus ponens ell c modus ponens ell e indeed  
  meta fx ++ meta fz == meta fy qed
end math ]"

"[  math in theory system Q lemma lemma negativeToRight(Less)(R) says
for all terms meta fx comma meta fy comma meta fz indeed
meta fx ++ -- meta fy << meta fz infer meta fx << meta fz ++ meta fy 
end lemma end math ]"

"[ math system Q proof of lemma negativeToRight(Less)(R) reads
any term meta fx comma meta fy comma meta fz end line
line ell a premise meta fx ++ -- meta fy << meta fz end line
line ell b because lemma lessAddition(R) modus ponens ell a indeed 
  meta fx ++ -- meta fy ++ meta fy << meta fz ++ meta fy end line

line ell c because lemma three2threeTerms(R) indeed 
  meta fx ++ -- meta fy ++ meta fy == meta fx ++ meta fy ++ -- meta fy end line
line ell d because lemma x=x+y-y(R) indeed meta fx == meta fx ++ meta fy ++ -- meta fy end line
line ell e because lemma ==Symmetry modus ponens ell d indeed 
  meta fx ++ meta fy ++ -- meta fy == meta fx end line
line ell f because lemma ==Transitivity modus ponens ell c modus ponens ell e indeed
  meta fx ++ -- meta fy ++ meta fy == meta fx end line

because lemma subLessLeft(R) modus ponens ell f modus ponens ell b indeed 
  meta fx << meta fz ++ meta fy qed
end math ]"


"[  math in theory system Q lemma lemma !!==Symmetry says
for all terms meta fx comma meta fy indeed
meta fx !!== meta fy infer meta fy !!== meta fx 
end lemma end math ]"

"[ math system Q proof of lemma !!==Symmetry reads
block any term meta fx comma meta fy end line
line ell a premise meta fy == meta fx end line
because lemma ==Symmetry modus ponens ell a indeed meta fx == meta fy end line
line ell b end block

any term meta fx comma meta fy end line
line ell c because 1rule deduction modus ponens ell b indeed 
  meta fy == meta fx imply meta fx == meta fy end line
line ell d premise meta fx !!== meta fy end line
because prop lemma mt modus ponens ell c modus ponens ell d indeed meta fy !!== meta fx qed
end math ]"


-------------(23.10.06)


"[  math in theory system Q lemma lemma negativeToRight(Eq)(R) says
for all terms meta fx comma meta fy comma meta fz indeed
meta fx ++ -- meta fy == meta fz infer meta fx == meta fz ++ meta fy
end lemma end math ]"

"[ math system Q proof of lemma negativeToRight(Eq)(R) reads
any term meta fx comma meta fy comma meta fz end line
line ell a premise meta fx ++ -- meta fy == meta fz end line
line ell b because 1rule adhoc eqAddition(R) modus ponens ell a indeed
  meta fx ++ -- meta fy ++ meta fy == meta fz ++ meta fy end line

line ell c because lemma x=x+y-y(R) indeed meta fx == meta fx ++ meta fy ++ -- meta fy end line

line ell e because lemma three2threeTerms(R) indeed
  meta fx ++ meta fy ++ -- meta fy == meta fx ++ -- meta fy ++ meta fy end line

because lemma ==Transitivity4 modus ponens ell c modus ponens ell e modus ponens ell b indeed 
  meta fx == meta fz ++ meta fy qed
end math ]"

"[  math in theory system Q lemma lemma negativeToRight(Eq)(1 term)(R) says
for all terms meta fx comma meta fy indeed
meta fx ++ -- meta fy == 00 infer meta fx == meta fy
end lemma end math ]"

"[ math system Q proof of lemma negativeToRight(Eq)(1 term)(R) reads
any term meta fx comma meta fy end line
line ell a premise meta fx ++ -- meta fy == 00 end line
line ell b because lemma negativeToRight(Eq)(R) modus ponens ell a indeed
  meta fx == 00 ++ meta fy end line

line ell c because lemma plus0Left(R) indeed 00 ++ meta fy == meta fy end line

because lemma ==Transitivity modus ponens ell b modus ponens ell c indeed meta fx == meta fy qed
end math ]"

"[  math in theory system Q lemma lemma doubleMinus(R) says
for all terms meta fx indeed -- -- meta fx == meta fx 
end lemma end math ]"

"[ math system Q proof of lemma doubleMinus(R) reads

any term meta fx end line
line ell a because lemma negative(R) indeed -- meta fx ++ -- -- meta fx == 00 end line

line ell b because lemma plusCommutativity(R) indeed 
  -- meta fx ++ -- -- meta fx ==  -- -- meta fx ++ -- meta fx end line
line ell c because lemma ==Symmetry modus ponens ell b indeed
  -- -- meta fx ++ -- meta fx == -- meta fx ++ -- -- meta fx end line

line ell d because lemma ==Transitivity modus ponens ell c modus ponens ell a indeed
  -- -- meta fx ++ -- meta fx == 00 end line
because lemma negativeToRight(Eq)(1 term)(R) modus ponens ell d indeed
  -- -- meta fx == meta fx qed
end math ]"


"[  math in theory system Q lemma lemma uniqueNegative(R) says
for all terms meta fx comma meta fy comma meta fz indeed
meta fx ++ meta fy == 00 infer meta fx ++ meta fz == 00 infer meta fy == meta fz
end lemma end math ]"

"[ math system Q proof of lemma uniqueNegative(R) reads
any term meta fx comma meta fy comma meta fz end line
line ell a premise meta fx ++ meta fy == 00 end line
line ell b premise meta fx ++ meta fz == 00 end line

line ell c because lemma plusCommutativity(R) indeed meta fy ++ meta fx == meta fx ++ meta fy end line
line ell d because lemma ==Transitivity modus ponens ell c modus ponens ell a indeed 
  meta fy ++ meta fx == 00 end line
line ell e because lemma positiveToRight(Eq)(R) modus ponens ell d indeed 
  meta fy == 00 ++ -- meta fx end line

line ell f because lemma plusCommutativity(R) indeed meta fz ++ meta fx == meta fx ++ meta fz end line
line ell g because lemma ==Transitivity modus ponens ell f modus ponens ell b indeed 
  meta fz ++ meta fx == 00 end line
line ell h because lemma positiveToRight(Eq)(R) modus ponens ell g indeed 
  meta fz == 00 ++ -- meta fx end line
line ell i because lemma ==Symmetry modus ponens ell h indeed 00 ++ -- meta fx == meta fz end line

because lemma ==Transitivity modus ponens ell e modus ponens ell i indeed 
 meta fy == meta fz qed
end math ]"


"[  math in theory system Q lemma lemma subtractEquationsLeft(R) says
for all terms meta fx comma meta fy comma meta fz comma meta fu indeed
meta fx ++ meta fz == meta fy ++ meta fu infer meta fx == meta fy infer meta fz == meta fu
end lemma end math ]"

"[ math system Q proof of lemma subtractEquationsLeft(R) reads
any term meta fx comma meta fy comma meta fz comma meta fu end line
line ell b premise meta fx ++ meta fz == meta fy ++ meta fu end line
line ell a premise meta fx == meta fy end line
line ell c because lemma plusCommutativity(R) indeed meta fz ++ meta fx == meta fx ++ meta fz end line
line ell d because lemma plusCommutativity(R) indeed meta fy ++ meta fu == meta fu ++ meta fy end line

line ell e because lemma ==Transitivity4 modus ponens ell c modus ponens ell b modus ponens ell d indeed
  meta fz ++ meta fx == meta fu ++ meta fy end line

because lemma subtractEquations(R) modus ponens ell e modus ponens ell a indeed meta fz == meta fu qed
end math ]"

"[  math in theory system Q lemma lemma eqNegated(R) says
for all terms meta fx comma meta fy indeed
meta fx == meta fy infer -- meta fx == -- meta fy 
end lemma end math ]"

"[ math system Q proof of lemma eqNegated(R) reads
any term meta fx comma meta fy end line
line ell a premise meta fx == meta fy end line
line ell b because lemma negative(R) indeed meta fx ++ -- meta fx == 00 end line
line ell c because lemma negative(R) indeed meta fy ++ -- meta fy == 00 end line
line ell d because lemma ==Symmetry modus ponens ell c indeed 00 == meta fy ++ -- meta fy end line
line ell e because lemma ==Transitivity modus ponens ell b modus ponens ell d indeed
  meta fx ++ -- meta fx == meta fy ++ -- meta fy end line
because lemma subtractEquationsLeft(R) modus ponens ell e modus ponens ell a indeed 
  -- meta fx == -- meta fy qed
end math ]"




"[ math in theory system Q lemma lemma neqNegated(R) says
for all terms meta fx comma meta fy indeed
meta fx !!== meta fy infer -- meta fx !!== -- meta fy
end lemma end math ]"

"[ math system Q proof of lemma neqNegated(R) reads
block any term meta fx comma meta fy end line
line ell big a premise meta fx !!== meta fy end line
line ell big b premise -- meta fx == -- meta fy end line
line ell big c because lemma eqNegated(R) modus ponens ell big b indeed 
  -- -- meta fx == -- -- meta fy end line
line ell big d because lemma doubleMinus(R) indeed -- -- meta fx == meta fx end line
line ell big e because lemma ==Symmetry modus ponens ell big d indeed meta fx == -- -- meta fx end line
line ell big f because lemma doubleMinus(R) indeed -- -- meta fy == meta fy end line
line ell big g because lemma ==Transitivity4 modus ponens ell big e modus ponens ell big c modus ponens ell big f indeed meta fx == meta fy end line
because prop lemma from contradiction modus ponens ell big g modus ponens ell big a indeed
  -- meta fx !!== -- meta fy end line
line ell big h end block 

any term meta fx comma meta fy end line
line ell a because 1rule deduction modus ponens ell big h indeed
  meta fx !!== meta fy imply -- meta fx == -- meta fy imply not0 -- meta fx == -- meta fy end line
line ell b premise meta fx !!== meta fy end line
line ell c because 1rule mp modus ponens ell a modus ponens ell b indeed
  -- meta fx == -- meta fy imply not0 -- meta fx == -- meta fy end line
because prop lemma imply negation modus ponens ell c indeed not0 -- meta fx == -- meta fy qed
end math ]"

-----------(23.10.06)

"[  math in theory system Q lemma lemma subLeqRight(R) says
for all terms meta fx comma meta fy comma meta fz indeed
meta fx == meta fy infer meta fz <<== meta fx infer meta fz <<== meta fy
end lemma end math ]"

"[ math system Q proof of lemma subLeqRight(R) reads
any term meta fx comma meta fy comma meta fz end line
line ell a premise meta fx == meta fy end line
line ell b premise meta fz <<== meta fx end line
line ell c because lemma eqLeq(R) modus ponens ell a indeed meta fx <<== meta fy end line
because lemma leqTransitivity(R) modus ponens ell b modus ponens ell c indeed 
  meta fz <<== meta fy qed
end math ]"

"[  math in theory system Q lemma lemma leqNegated(R) says
for all terms meta fx comma meta fy indeed
meta fx <<== meta fy infer -- meta fy <<== -- meta fx
end lemma end math ]"

"[ math system Q proof of lemma leqNegated(R) reads
any term meta fx comma meta fy end line
line ell a premise meta fx <<== meta fy end line
line ell b because lemma leqAddition(R) modus ponens ell a indeed 
  meta fx ++ -- meta fx <<== meta fy ++ -- meta fx end line
line ell c because lemma negative(R) indeed meta fx ++ -- meta fx == 00 end line
line ell d because lemma subLeqLeft(R) modus ponens ell c modus ponens ell b indeed 
  00 <<== meta fy ++ -- meta fx end line
line ell e because lemma plusCommutativity(R) indeed
  meta fy ++ -- meta fx == -- meta fx ++ meta fy end line
line ell f because lemma subLeqRight(R) modus ponens ell e modus ponens ell d indeed
  00 <<== -- meta fx ++ meta fy end line
line ell g because lemma leqAddition(R) modus ponens ell f indeed
  00 ++ -- meta fy <<== -- meta fx ++ meta fy ++ -- meta fy end line

line ell h because lemma plus0Left(R) indeed 00 ++ -- meta fy == -- meta fy end line
line ell i because lemma x=x+y-y(R) indeed -- meta fx == -- meta fx ++ meta fy ++ -- meta fy end line
line ell j because lemma ==Symmetry modus ponens ell i indeed 
  -- meta fx ++ meta fy ++ -- meta fy == -- meta fx end line

line ell k because lemma subLeqLeft(R) modus ponens ell h modus ponens ell g indeed 
  -- meta fy <<== -- meta fx ++ meta fy ++ -- meta fy end line
because lemma subLeqRight(R) modus ponens ell j modus ponens ell k indeed 
  -- meta fy <<== -- meta fx qed
end math ]"

"[  math in theory system Q lemma lemma lessNegated(R) says
for all terms meta fx comma meta fy indeed
meta fx << meta fy infer -- meta fy << -- meta fx
end lemma end math ]"


"[ math system Q proof of lemma lessNegated(R) reads
any term meta fx comma meta fy end line
line ell a premise meta fx << meta fy end line
line ell b because lemma lessLeq(R) modus ponens ell a indeed 
  meta fx <<== meta fy end line
line ell c because lemma leqNegated(R) modus ponens ell b indeed
  -- meta fy <<== -- meta fx end line

line ell d because lemma lessNeq(R) modus ponens ell a indeed
  meta fx !!== meta fy end line
line ell e because lemma neqNegated(R) modus ponens ell d indeed
  -- meta fx !!== -- meta fy end line
line ell f because lemma !!==Symmetry modus ponens ell e indeed
  -- meta fy !!== -- meta fx end line

because lemma leqNeqLess(R) modus ponens ell c modus ponens ell f indeed 
  -- meta fy << -- meta fx qed
end math ]"


"[  math in theory system Q lemma lemma -0=0(R) says -- 00 == 00
end lemma end math ]"

"[ math system Q proof of lemma -0=0(R) reads
line ell a because lemma negative(R) indeed 00 ++ -- 00 == 00 end line
line ell b because lemma plus0(R) indeed 00 ++ 00 == 00 end line
because lemma uniqueNegative(R) modus ponens ell a modus ponens ell b indeed -- 00 == 00 qed
end math ]"

"[  math in theory system Q lemma lemma negativeNegated(R) says
for all terms meta fx indeed
meta fx << 00 infer 00 << -- meta fx 
end lemma end math ]"

"[ math system Q proof of lemma negativeNegated(R) reads
any term meta fx end line
line ell a premise meta fx << 00 end line
line ell b because lemma lessNegated(R) modus ponens ell a indeed -- 00 << -- meta fx end line
line ell c because lemma -0=0(R) indeed -- 00 == 00 end line
because lemma subLessLeft(R) modus ponens ell c modus ponens ell b indeed 00 << -- meta fx qed
end math ]"

"[  math in theory system Q lemma lemma from leqGeq(R) says
for all terms meta a comma meta fx comma meta fy indeed
meta fx <<== meta fy imply meta a infer
meta fy <<== meta fx imply meta a infer meta a
end lemma end math ]"

"[ math system Q proof of lemma from leqGeq(R) reads
any term meta a comma meta fx comma meta fy end line
line ell a premise meta fx <<== meta fy imply meta a end line
line ell b premise meta fy <<== meta fx imply meta a end line
line ell c because lemma leqTotality(R) indeed meta fx <<== meta fy or0 meta fy <<== meta fx end line
because prop lemma from disjuncts modus ponens ell c modus ponens ell a modus ponens ell b indeed meta a qed
end math ]"

--------(24.10.06)

"[  math in theory system Q lemma lemma fromLess(R) says
for all terms meta fx comma meta fy indeed
meta fx << meta fy infer not0 meta fy <<== meta fx
end lemma end math ]"

"[ math system Q proof of lemma fromLess(R) reads
block any term meta fx comma meta fy end line
line ell a premise meta fx << meta fy end line
line ell b premise meta fy <<== meta fx end line
line ell c because lemma lessLeq(R) modus ponens ell a indeed meta fx <<== meta fy end line
line ell d because lemma leqAntisymmetry(R) modus ponens ell c modus ponens ell b indeed
  meta fx == meta fy end line
line ell e because lemma lessNeq(R) modus ponens ell a indeed meta fx !!== meta fy end line
because prop lemma from contradiction modus ponens ell d modus ponens ell e indeed 
  not0 meta fy <<== meta fx end line
line ell big a end block

any term meta fx comma meta fy end line

line ell a because 1rule deduction modus ponens ell big a indeed
  meta fx << meta fy imply meta fy <<== meta fx imply 
  not0 meta fy <<== meta fx end line
line ell b premise meta fx << meta fy end line  
line ell c because 1rule mp modus ponens ell a modus ponens ell b indeed 
  meta fy <<== meta fx imply not0 meta fy <<== meta fx end line

because prop lemma imply negation modus ponens ell c indeed not0 meta fy <<== meta fx qed
end math ]"


"[  math in theory system Q lemma lemma nonnegativeNumerical(R) says
for all terms meta fx indeed
00 <<== meta fx infer |r meta fx | == meta fx
end lemma end math ]"

"[ math system Q proof of lemma nonnegativeNumerical(R) reads
any term meta fx end line
line ell a premise 00 <<== meta fx end line
line ell b because 1rule ifThenElse true modus ponens ell a indeed
  if( 00 <<== meta fx , meta fx , -- meta fx ) == meta fx end line
because 1rule repetition modus ponens ell b indeed |r meta fx | == meta fx qed ""; XXREP |r |
end math ]"

"[  math in theory system Q lemma lemma negativeNumerical(R) says
for all terms meta fx indeed
meta fx << 00 infer |r meta fx | == -- meta fx
end lemma end math ]"


"[ math system Q proof of lemma negativeNumerical(R) reads
any term meta fx end line
line ell a premise meta fx << 00 end line
line ell b because lemma fromLess(R) modus ponens ell a indeed not0 00 <<== meta fx end line
line ell c because 1rule ifThenElse false modus ponens ell b indeed
  if( 00 <<== meta fx , meta fx , -- meta fx ) == -- meta fx end line
because 1rule repetition modus ponens ell c indeed |r meta fx | == -- meta fx qed ""; XXREP |r |
end math ]"


"[  math in theory system Q lemma lemma 0<=|x|(R) says
for all terms meta fx indeed
00 <<== |r meta fx |
end lemma end math ]"

"[ math system Q proof of lemma 0<=|x|(R) reads
block any term meta fx end line
line ell a premise 00 <<== meta fx end line
line ell b because lemma nonnegativeNumerical(R) modus ponens ell a indeed |r meta fx | == meta fx end line
line ell c because lemma ==Symmetry modus ponens ell b indeed meta fx == |r meta fx | end line
because lemma subLeqRight(R) modus ponens ell c modus ponens ell a indeed 00 <<== |r meta fx | end line
line ell d end block 

block any term meta fx end line
line ell e premise not0 00 <<== meta fx end line
line ell f because lemma toLess(R) modus ponens ell e indeed meta fx << 00 end line
line ell g because lemma negativeNumerical(R) modus ponens ell f indeed |r meta fx | == -- meta fx end line
line ell h because lemma ==Symmetry modus ponens ell g indeed -- meta fx == |r meta fx | end line
line ell i because lemma negativeNegated(R) modus ponens ell f indeed 00 << -- meta fx end line
line ell j because lemma lessLeq(R) modus ponens ell i indeed 00 <<== -- meta fx end line
because lemma subLeqRight(R) modus ponens ell h modus ponens ell j indeed 00 <<== |r meta fx | end line
line ell k end block

any term meta fx end line
line ell l because 1rule deduction modus ponens ell d indeed 00 <<== meta fx imply 00 <<== |r meta fx | end line
line ell m because 1rule deduction modus ponens ell k indeed not0 00 <<== meta fx imply 00 <<== |r meta fx | end line

because prop lemma from negations modus ponens ell l modus ponens ell m indeed 00 <<== |r meta fx | qed
end math ]"

-----------(24.10.06)

"[  math in theory system Q lemma lemma positiveNegated(R) says
for all terms meta fx indeed
00 << meta fx infer -- meta fx << 00
end lemma end math ]"

"[ math system Q proof of lemma positiveNegated(R) reads
any term meta fx end line
line ell a premise 00 << meta fx end line
line ell b because lemma lessNegated(R) modus ponens ell a indeed
  -- meta fx << -- 00 end line
line ell c because lemma -0=0(R) indeed -- 00 == 00 end line

because lemma subLessRight(R) modus ponens ell c modus ponens ell b indeed -- meta fx << 00 qed
end math ]"


"[  math in theory system Q lemma lemma addEquations(R) says
for all terms meta fx comma meta fy comma meta fz comma meta fu indeed
meta fx == meta fy infer meta fz == meta fu infer meta fx ++ meta fz == meta fy ++ meta fu
end lemma end math ]"

"[ math system Q proof of lemma addEquations(R) reads
any term meta fx comma meta fy comma meta fz comma meta fu end line
line ell a premise meta fx == meta fy end line
line ell b premise meta fz == meta fu end line
line ell c because 1rule adhoc eqAddition(R) modus ponens ell a indeed 
  meta fx ++ meta fz == meta fy ++ meta fz end line
line ell d because lemma eqAdditionLeft(R) modus ponens ell b indeed
  meta fy ++ meta fz == meta fy ++ meta fu end line
because lemma ==Transitivity modus ponens ell c modus ponens ell d indeed 
  meta fx ++ meta fz == meta fy ++ meta fu qed 
end math ]"


"[  math in theory system Q lemma lemma distributionOut(R) says
for all terms meta fx comma meta fy comma meta fz indeed
meta fx ** meta fy ++ meta fx ** meta fz == meta fx ** ( meta fy ++ meta fz )
end lemma end math ]"

"[ math system Q proof of lemma distributionOut(R) reads
any term meta fx comma meta fy comma meta fz end line
line ell a because lemma distribution(R) indeed 
  meta fx ** ( meta fy ++ meta fz ) == meta fx ** meta fy ++ meta fx ** meta fz  end line
because lemma ==Symmetry modus ponens ell a indeed 
  meta fx ** meta fy ++ meta fx ** meta fz == meta fx ** ( meta fy ++ meta fz ) qed
end math ]"


"[  math in theory system Q lemma lemma ==Transitivity5 says
for all terms meta fx comma meta fy comma meta fz comma meta fu comma meta fv indeed
meta fx == meta fy infer meta fy == meta fz infer meta fz == meta fu infer
meta fu == meta fv infer meta fx == meta fv
end lemma end math ]"

"[ math system Q proof of lemma ==Transitivity5 reads
any term meta fx comma meta fy comma meta fz comma meta fu comma meta fv end line
line ell a premise meta fx == meta fy end line
line ell b premise meta fy == meta fz end line
line ell c premise meta fz == meta fu end line
line ell d premise meta fu == meta fv end line
line ell e because lemma ==Transitivity4 modus ponens ell a modus ponens ell b modus ponens ell c indeed
  meta fx == meta fu end line
because lemma ==Transitivity modus ponens ell e modus ponens ell d indeed 
meta fx == meta fv qed
end math ]"


"[  math in theory system Q lemma lemma  x*0+x=x(R) says
for all terms meta fx indeed 
meta fx ** 00 ++ meta fx == meta fx 
end lemma end math ]"

"[ math system Q proof of lemma x*0+x=x(R) reads
any term meta fx end line
line ell big a because lemma times1(R) indeed meta fx ** 01 ==  meta fx end line
line ell a because lemma ==Symmetry modus ponens ell big a indeed meta fx == meta fx ** 01 end line
line ell b because lemma eqAdditionLeft(R) modus ponens ell a indeed 
  meta fx ** 00 ++ meta fx == meta fx ** 00 ++ meta fx ** 01 end line

line ell c because lemma distribution(R) indeed 
  meta fx ** ( 00 ++ 01 ) == meta fx ** 00 ++ meta fx ** 01 end line
line ell d because lemma ==Symmetry modus ponens ell c indeed 
  meta fx ** 00 ++ meta fx ** 01 == meta fx ** ( 00 ++ 01 ) end line

line ell e because lemma plus0Left(R) indeed 00 ++ 01 == 01 end line
line ell f because lemma eqMultiplicationLeft(R) modus ponens ell e indeed 
  meta fx ** ( 00 ++ 01 ) == meta fx ** 01 end line

because lemma ==Transitivity5 modus ponens ell b modus ponens ell d modus ponens ell f 
  modus ponens ell big a indeed  meta fx ** 00 ++ meta fx == meta fx qed
end math ]"

--------(24.10.06)




"[  math in theory system Q lemma lemma x*0=0(R)fff says
for all terms meta fx indeed
meta fx ** 00 == 00
end lemma end math ]"

"[ math system Q proof of lemma x*0=0(R)fff reads
any term meta fx end line
line ell a because lemma x=x+(y-y)(R) indeed 
   meta fx ** 00 == meta fx ** 00 ++ ( meta fx ++ -- meta fx ) end line

line ell b because lemma plusAssociativity(R) indeed 
  meta fx ** 00 ++ meta fx ++ -- meta fx == 
  meta fx ** 00 ++ ( meta fx ++ -- meta fx ) end line
line ell c because lemma ==Symmetry modus ponens ell b indeed
  meta fx ** 00 ++ ( meta fx ++ -- meta fx ) == meta fx ** 00 ++ meta fx ++ -- meta fx end line

line ell d because lemma x*0+x=x(R) indeed meta fx ** 00 ++ meta fx == meta fx end line
line ell e because 1rule adhoc eqAddition(R) modus ponens ell d indeed
  meta fx ** 00 ++ meta fx ++ -- meta fx == meta fx ++ -- meta fx end line

line ell f because lemma negative(R) indeed meta fx ++ -- meta fx == 00 end line

because lemma ==Transitivity5 modus ponens ell a modus ponens ell c modus ponens ell e modus ponens ell f   
  indeed meta fx ** 00 == 00 qed
end math ]"


"[  math in theory system Q lemma lemma times(-1)(R) says
for all terms meta fx indeed
meta fx ** (--01) == -- meta fx 
end lemma end math ]"

"[ math system Q proof of lemma times(-1)(R) reads
any term meta fx end line
line ell a because lemma negative(R) indeed 01 ++ (--01) == 00 end line
line ell b because lemma plusCommutativity(R) indeed (--01) ++ 01 == 01 ++ (--01) end line
line ell c because lemma ==Transitivity modus ponens ell b modus ponens ell a indeed 
  (--01) ++ 01 == 00 end line
line ell d because lemma eqMultiplicationLeft(R) modus ponens ell c indeed 
  meta fx ** ( (--01) ++ 01 ) == meta fx ** 00 end line
line ell e because lemma x*0=0(R)fff indeed meta fx ** 00 == 00 end line
line ell f because lemma ==Transitivity modus ponens ell d modus ponens ell e indeed
  meta fx ** ( (--01) ++ 01 ) == 00 end line



line ell g because lemma distribution(R) indeed 
  meta fx ** ( (--01) ++ 01 ) == meta fx ** (--01) ++ meta fx ** 01 end line
line ell h because lemma ==Symmetry modus ponens ell g indeed
  meta fx ** (--01) ++ meta fx ** 01 ==  meta fx ** ( (--01) ++ 01 ) end line

line ell i because lemma ==Transitivity modus ponens ell h modus ponens ell f indeed
  meta fx ** (--01) ++ meta fx ** 01 == 00 end line
line ell j because lemma positiveToRight(Eq)(R) modus ponens ell i indeed
  meta fx ** (--01) == 00 ++ -- ( meta fx ** 01 ) end line

line ell k because lemma plus0Left(R) indeed 
  00 ++ -- ( meta fx ** 01 ) == -- ( meta fx ** 01 ) end line

line ell m because lemma ==Transitivity modus ponens ell j modus ponens ell k indeed
  meta fx ** (--01) == -- ( meta fx ** 01 ) end line

line ell n because lemma times1(R) indeed meta fx ** 01 == meta fx end line
line ell o because lemma eqNegated(R) modus ponens ell n indeed 
  -- ( meta fx ** 01 ) == -- meta fx end line

because lemma ==Transitivity modus ponens ell m modus ponens ell o indeed  
  meta fx ** (--01) == -- meta fx qed
end math ]"


"[  math in theory system Q lemma lemma times(-1)Left(R) says
for all terms meta fx indeed
(--01) ** meta fx == -- meta fx
end lemma end math ]"

"[ math system Q proof of lemma times(-1)Left(R) reads
any term meta fx end line
line ell a because lemma times(-1)(R) indeed meta fx ** (--01) == -- meta fx end line
line ell b because lemma timesCommutativity(R) indeed (--01) ** meta fx == meta fx ** (--01) end line
because lemma ==Transitivity modus ponens ell b modus ponens ell a indeed 
  (--01) ** meta fx == -- meta fx qed
end math ]"



(************)

"[  math in theory system Q lemma lemma -x-y=-(x+y)(R) says
for all terms meta fx comma meta fy indeed
-- meta fx ++ -- meta fy == -- ( meta fx ++ meta fy ) 
end lemma end math ]"


"[ math system Q proof of lemma -x-y=-(x+y)(R) reads
any term meta fx comma meta fy end line

line ell a because lemma times(-1)Left(R) indeed (--01) ** meta fx == -- meta fx end line
line ell b because lemma times(-1)Left(R) indeed (--01) ** meta fy == -- meta fy end line
line ell c because lemma addEquations(R) modus ponens ell a modus ponens ell b indeed
  (--01) ** meta fx ++ (--01) ** meta fy == -- meta fx ++ -- meta fy end line
line ell d because lemma ==Symmetry modus ponens ell c indeed
 -- meta fx ++ -- meta fy == (--01) ** meta fx ++ (--01) ** meta fy end line

line ell e because lemma distributionOut(R) indeed
  (--01) ** meta fx ++ (--01) ** meta fy == (--01) ** ( meta fx ++ meta fy ) end line

line ell f because lemma times(-1)Left(R) indeed
  (--01) ** ( meta fx ++ meta fy ) == -- ( meta fx ++ meta fy ) end line

because lemma ==Transitivity4 modus ponens ell d modus ponens ell e modus ponens ell f 
  indeed -- meta fx ++ -- meta fy == -- ( meta fx ++ meta fy ) qed
end math ]"


"[  math in theory system Q lemma lemma lessTotality(R) says
for all terms meta fx comma meta fy indeed
meta fx << meta fy or0 meta fx == meta fy or0 meta fy << meta fx
end lemma end math ]"

"[ math system Q proof of lemma lessTotality(R) reads
block any term meta fx comma meta fy end line
line ell a premise not0 meta fx << meta fy end line
line ell b premise meta fx !!== meta fy end line
line ell c because lemma fromNotLess(R) modus ponens ell a indeed 
  meta fy <<== meta fx end line
line ell d because lemma !!==Symmetry modus ponens ell b indeed
  meta fy !!== meta fx end line
because lemma leqNeqLess(R) modus ponens ell c  modus ponens ell d indeed meta fy << meta fx end line
line ell e end block

any term meta fx comma meta fy end line
line ell f because 1rule deduction modus ponens ell e indeed 
  not0 meta fx << meta fy imply meta fx !!== meta fy imply meta fy << meta fx end line
because 1rule repetition modus ponens ell f indeed 
  meta fx << meta fy or0 meta fx == meta fy or0 meta fy << meta fx qed ""; XXREP or
  
end math ]"


"[  math in theory system Q lemma lemma sameNumerical(R) says
for all terms meta fx comma meta fy indeed
meta fx == meta fy infer |r meta fx | == |r meta fy |
end lemma end math ]"

"[ math system Q proof of lemma sameNumerical(R) reads
block any term meta fx comma meta fy end line
line ell big a premise 00 <<== meta fx end line
line ell big b premise meta fx == meta fy end line
line ell big c because lemma nonnegativeNumerical(R) modus ponens ell big a indeed
  |r meta fx | == meta fx end line

line ell big d because lemma subLeqRight(R) modus ponens ell big b modus ponens ell big a indeed  
  00 <<== meta fy end line
line ell big e because lemma nonnegativeNumerical(R) modus ponens ell big d indeed

  |r meta fy | == meta fy end line
line ell big f because lemma ==Symmetry modus ponens ell big e indeed meta fy == |r meta fy | end line

because lemma ==Transitivity4 modus ponens ell big c modus ponens ell big b modus ponens ell 
  big f indeed |r meta fx | == |r meta fy | end line
line ell big g end block
	

block any term meta fx comma meta fy end line
line ell d premise not0 00 <<== meta fx end line
line ell e premise meta fx == meta fy end line

line ell f because lemma toLess(R) modus ponens ell d indeed meta fx << 00 end line
line ell g because lemma negativeNumerical(R) modus ponens ell f indeed
  |r meta fx | == -- meta fx end line

line ell h because lemma subLessLeft(R) modus ponens ell e modus ponens ell f indeed meta fy << 00 end line
line ell i because lemma negativeNumerical(R) modus ponens ell h indeed
  |r meta fy | == -- meta fy end line
line ell j because lemma ==Symmetry modus ponens ell i indeed -- meta fy == |r meta fy | end line

line ell k because lemma eqNegated(R) modus ponens ell e indeed -- meta fx == -- meta fy end line
because lemma ==Transitivity4 modus ponens ell g modus ponens ell k modus ponens ell j indeed 
  |r meta fx | == |r meta fy | end line
line ell l end block

any term meta fx comma meta fy end line
line ell m premise meta fx == meta fy end line
line ell n because 1rule deduction modus ponens ell big g indeed
  00 <<== meta fx imply meta fx == meta fy imply |r meta fx | == |r meta fy | end line
line ell o because 1rule deduction modus ponens ell l indeed
  not0 00 <<== meta fx imply meta fx == meta fy imply |r meta fx | == |r meta fy | end line
line ell p because prop lemma from negations modus ponens ell n modus ponens ell o indeed
  meta fx == meta fy imply |r meta fx | == |r meta fy | end line

because 1rule mp modus ponens ell p modus ponens ell m indeed |r meta fx | == |r meta fy | qed
end math ]"


"[  math in theory system Q lemma lemma minusNegated(R) says
for all terms meta fx comma meta fy indeed
-- ( meta fx ++ -- meta fy ) == meta fy ++ -- meta fx
end lemma end math ]"

"[ math system Q proof of lemma minusNegated(R) reads
any term meta fx comma meta fy end line
line ell a because lemma doubleMinus(R) indeed -- -- meta fy == meta fy end line
line ell b because 1rule adhoc eqAddition(R) modus ponens ell a indeed
   -- -- meta fy ++ -- meta fx == meta fy ++ -- meta fx end line
line ell c because lemma ==Symmetry modus ponens ell b indeed 
  meta fy ++ -- meta fx == -- -- meta fy ++ -- meta fx end line

line ell d because lemma -x-y=-(x+y)(R) indeed
  -- -- meta fy ++ -- meta fx == -- ( -- meta fy ++ meta fx ) end line

line ell e because lemma plusCommutativity(R) indeed -- meta fy ++ meta fx == meta fx ++ -- meta fy end line
line ell f because lemma eqNegated(R) modus ponens ell e indeed
  -- ( -- meta fy ++ meta fx ) == -- ( meta fx ++ -- meta fy ) end line

line ell g because lemma ==Transitivity4 modus ponens ell c modus ponens ell d modus ponens ell f indeed
  meta fy ++ -- meta fx == -- ( meta fx ++ -- meta fy ) end line

because lemma ==Symmetry modus ponens ell g indeed 
  -- ( meta fx ++ -- meta fy ) == meta fy ++ -- meta fx qed

end math ]"


-----------(24.10.06)

"[  math in theory system Q lemma lemma positiveNumerical(R) says
for all terms meta fx indeed
00 << meta fx infer |r meta fx | == meta fx
end lemma end math ]"

"[  math system Q proof of lemma positiveNumerical(R) reads
any term meta fx end line
line ell a premise 00 << meta fx end line
line ell b because lemma lessLeq(R) modus ponens ell a indeed 00 <<== meta fx end line
because lemma nonnegativeNumerical(R) modus ponens ell b indeed |r meta fx | == meta fx qed
end math ]"


"[  math in theory system Q lemma lemma signNumerical(+)(R) says
for all terms meta fx indeed
00 << meta fx infer |r meta fx | == |r -- meta fx |
end lemma end math ]"


"[ math system Q proof of lemma signNumerical(+)(R) reads
any term meta fx end line
line ell a premise 00 << meta fx end line
line ell b because lemma positiveNumerical(R) modus ponens ell a indeed 
  |r meta fx | == meta fx end line
line ell c because lemma positiveNegated(R) modus ponens ell a indeed -- meta fx << 00 end line
line ell d because lemma negativeNumerical(R) modus ponens ell c indeed
  |r -- meta fx | == -- -- meta fx end line
line ell e because lemma doubleMinus(R) indeed -- -- meta fx == meta fx end line
line ell f because lemma ==Transitivity modus ponens ell d modus ponens ell e indeed
  |r -- meta fx | == meta fx end line
line ell g because lemma ==Symmetry modus ponens ell f indeed 
  meta fx == |r -- meta fx | end line

because lemma ==Transitivity modus ponens ell b modus ponens ell g indeed 
  |r meta fx | == |r -- meta fx | qed
end math ]"



"[  math in theory system Q lemma lemma signNumerical(R) says
for all terms meta fx indeed
|r meta fx | == |r -- meta fx |
end lemma end math ]"

"[ math system Q proof of lemma signNumerical(R) reads

block any term meta fx end line
line ell a premise meta fx << 00 end line
line ell b because lemma negativeNegated(R) modus ponens ell a indeed 00 << -- meta fx end line
line ell c because lemma signNumerical(+)(R) modus ponens ell b indeed
  |r -- meta fx | == |r -- -- meta fx | end line
line ell d because lemma doubleMinus(R) indeed -- -- meta fx == meta fx end line
line ell e because lemma sameNumerical(R) modus ponens ell d indeed 
  |r -- -- meta fx | == |r meta fx | end line
line ell f because lemma ==Transitivity modus ponens ell c modus ponens ell e indeed
  |r -- meta fx | == |r meta fx | end line
because lemma ==Symmetry modus ponens ell f indeed |r meta fx | == |r -- meta fx | end line
line ell big a end block


block any term meta fx end line
line ell a premise meta fx == 00 end line

line ell b because lemma eqNegated(R) modus ponens ell a indeed -- meta fx == -- 00 end line
line ell c because lemma -0=0(R) indeed -- 00 == 00 end line
line ell d because lemma ==Symmetry modus ponens ell a indeed 00 == meta fx end line

line ell e because lemma ==Transitivity4 modus ponens ell b modus ponens ell c 
  modus ponens ell d indeed -- meta fx == meta fx end line

line ell f because lemma ==Symmetry modus ponens ell e indeed  meta fx == -- meta fx end line
because lemma sameNumerical(R) modus ponens ell f indeed |r meta fx | == |r -- meta fx | end line
line ell big b end block


block any term meta fx end line
line ell a premise  00 << meta fx end line
because lemma signNumerical(+)(R) modus ponens ell a indeed 
  |r meta fx | == |r -- meta fx | end line
line ell big c end block

any term meta fx end line
line ell big d because 1rule deduction modus ponens ell big a indeed
  meta fx << 00 imply |r meta fx | == |r -- meta fx | end line 


line ell big e because 1rule deduction modus ponens ell big b indeed
  meta fx == 00 imply |r meta fx | == |r -- meta fx | end line
line ell big f because 1rule deduction modus ponens ell big c indeed
  00 << meta fx imply |r meta fx | == |r -- meta fx | end line

line ell big g because lemma lessTotality(R) indeed
  meta fx << 00 or0 meta fx == 00 or0 00 << meta fx end line

because prop lemma from three disjuncts modus ponens ell big g 
  modus ponens ell big d modus ponens ell big e modus ponens ell big f indeed 
  |r meta fx | == |r -- meta fx | qed
end math ]"

"[  math in theory system Q lemma lemma numericalDifference(R) says
for all terms meta fx comma meta fy indeed
|r meta fx ++ -- meta fy | == |r meta fy ++ -- meta fx |
end lemma end math ]"

"[ math system Q proof of lemma numericalDifference(R) reads
any term meta fx comma meta fy end line
line ell a because lemma signNumerical(R) indeed 
  |r meta fx ++ -- meta fy | == |r -- ( meta fx ++ -- meta fy ) | end line

line ell b because lemma minusNegated(R) indeed
  -- ( meta fx ++ -- meta fy ) == meta fy ++ -- meta fx end line
line ell c because lemma sameNumerical(R) modus ponens ell b indeed
  |r -- ( meta fx ++ -- meta fy ) | == |r meta fy ++ -- meta fx | end line
  
because lemma ==Transitivity modus ponens ell a modus ponens ell c indeed 
|r meta fx ++ -- meta fy | == |r meta fy ++ -- meta fx | qed
end math ]"

--------(25.10.06)

"[  math in theory system Q lemma lemma x<=|x|(R) says
for all terms meta fx indeed
meta fx <<== |r meta fx |
end lemma end math ]"


"[ math system Q proof of lemma x<=|x|(R) reads
block any term meta fx end line
line ell a premise 00 <<== meta fx end line
line ell b because lemma nonnegativeNumerical(R) indeed |r meta fx | == meta fx end line
line ell c because lemma ==Symmetry modus ponens ell b indeed meta fx == |r meta fx | end line
because lemma eqLeq(R) modus ponens ell c indeed meta fx <<== |r meta fx | end line
line ell big a end block

block any term meta fx end line
line ell a premise meta fx <<== 00 end line
line ell b because lemma 0<=|x|(R) indeed 00 <<== |r meta fx | end line
because lemma leqTransitivity(R) modus ponens ell a modus ponens ell b indeed 
  meta fx <<== |r meta fx | end line
line ell big b end block

any term meta fx end line
line ell a because 1rule deduction modus ponens ell big a indeed 
  00 <<== meta fx imply meta fx <<== |r meta fx | end line
line ell b because 1rule deduction modus ponens ell big b indeed 
  meta fx <<== 00 imply meta fx <<== |r meta fx | end line

because lemma from leqGeq(R) modus ponens ell a modus ponens ell b indeed meta fx <<== |r meta fx | qed
end math ]"



"[  math in theory system Q lemma lemma USlimitIsUpperBound helper says
""; XX meta fy og meta fx er byttet om i forhold til hovedlemmaet
for all terms meta m comma meta n comma meta fep comma meta fx comma meta fy indeed
meta fx in0 setOfFxs imply
for all meta n indeed ( 00 << meta fep imply meta m <= meta n imply 
  |r meta fy ++ -- [ usF ; meta n ] | << meta fep ) imply
meta fx <<== meta fy 
end lemma end math ]"


"[ math system Q proof of lemma USlimitIsUpperBound helper reads 

block any term meta m comma meta n comma meta fep comma meta fx comma meta fy end line

block any term meta m comma meta n comma meta fep comma meta fx comma meta fy end line
line ell big a  premise meta fx in0 setOfFxs end line
line ell a premise for all meta n indeed ( 00 << meta fep imply meta m <= meta n imply 
  |r meta fy ++ -- [ usF ; meta n ] | << meta fep ) end line
line ell b premise 00 << meta fep end line

line ell big c because lemma a4 at meta m modus ponens ell a indeed
  00 << meta fep imply meta m <= meta m imply 
  |r meta fy ++ -- [ usF ; meta m ] | << meta fep end line

line ell c because axiom leqReflexivity indeed meta m <= meta m end line

line ell d because prop lemma mp2 modus ponens ell big c modus ponens ell b modus ponens ell c indeed
  |r meta fy ++ -- [ usF ; meta m ] | << meta fep end line

line ell e because lemma numericalDifference(R) indeed
  |r meta fy ++ -- [ usF ; meta m ] | == |r [ usF ; meta m ] ++ -- meta fy | end line
line ell f because lemma subLessLeft(R) modus ponens ell e modus ponens ell d indeed
  |r [ usF ; meta m ] ++ -- meta fy | << meta fep end line

line ell g because lemma x<=|x|(R) indeed
  [ usF ; meta m ] ++ -- meta fy <<== |r [ usF ; meta m ] ++ -- meta fy | end line
line ell h because lemma leqLessTransitivity(R) modus ponens ell g modus ponens ell f indeed
  [ usF ; meta m ] ++ -- meta fy << meta fep end line
line ell i because lemma negativeToRight(Less)(R) modus ponens ell h indeed
  [ usF ; meta m ] << meta fep ++ meta fy end line
line ell j because lemma plusCommutativity(R) indeed 
  meta fep ++ meta fy == meta fy ++ meta fep end line
line ell k because lemma subLessRight(R) modus ponens ell j modus ponens ell i indeed
  [ usF ; meta m ] << meta fy ++ meta fep end line

line ell l because axiom USisUpperBound indeed upperBound( [ usF ; meta m ] , setOfFxs ) end line
line ell m because 1rule fromUpperBound modus ponens ell l modus ponens ell big a indeed
   meta fx <<== [ usF ; meta m ] end line

line ell n because lemma leqLessTransitivity(R) modus ponens ell m modus ponens ell k indeed
  meta fx << meta fy ++ meta fep end line
line ell o because lemma lessNeq(R) modus ponens ell n indeed
  meta fx !!== meta fy ++ meta fep end line
because lemma !!==Symmetry modus ponens ell o indeed
  meta fy ++ meta fep !!== meta fx end line
line ell big b end block

line ell big q because 1rule deduction modus ponens ell big b indeed
meta fx in0 setOfFxs imply
for all meta n indeed ( 00 << meta fep imply meta m <= meta n imply 
  |r meta fy ++ -- [ usF ; meta n ] | << meta fep ) imply
00 << meta fep imply
meta fy ++ meta fep !!== meta fx end line

line ell big r premise meta fx in0 setOfFxs end line
line ell big s premise for all meta n indeed ( 00 << meta fep imply meta m <= meta n imply 
  |r meta fy ++ -- [ usF ; meta n ] | << meta fep ) end line



line ell big t because prop lemma mp2 modus ponens ell big q modus ponens ell big r 
  modus ponens ell big s indeed 00 << meta fep imply meta fy ++ meta fep !!== meta fx end line

because lemma toLeq(Advanced)(R) modus ponens ell big t indeed
  meta fx <<== meta fy end line
line ell big m end block

any term meta m comma meta n comma meta fep comma meta fx comma meta fy end line

because 1rule deduction modus ponens ell big m indeed
meta fx in0 setOfFxs imply
for all meta n indeed ( 00 << meta fep imply meta m <= meta n imply 
  |r meta fy ++ -- [ usF ; meta n ] | << meta fep ) imply
meta fx <<== meta fy qed
end math ]"


"[  math in theory system Q lemma lemma USlimitIsUpperBound says
for all terms meta m comma meta n comma meta fep comma meta fx comma meta fy indeed ""; XX ekstra variable
limit( meta fx , usF ) infer upperBound( meta fx , setOfFxs )
end lemma end math ]"

"[ math system Q proof of lemma USlimitIsUpperBound reads
block any term meta m comma meta n comma meta fep comma meta fx comma meta fy end line
line ell a premise limit( meta fx , usF ) end line
line ell b premise meta fy in0 setOfFxs end line
line ell c because 1rule fromLimit modus ponens ell a indeed
  for all meta fep indeed exist0 meta m indeed for all meta n indeed 
  ( 00 << meta fep imply meta m <= meta n imply 
     |r meta fx ++ -- [ usF ; meta n ] | << meta fep ) end line


line ell d because lemma a4 at meta ep modus ponens ell c indeed
 exist0 meta m indeed for all meta n indeed ( 00 << meta fep imply meta m <= meta n imply 
     |r meta fx ++ -- [ usF ; meta n ] | << meta fep )  end line

line ell e because lemma USlimitIsUpperBound helper indeed
meta fy in0 setOfFxs imply
for all meta n indeed ( 00 << meta fep imply meta m <= meta n imply 
  |r meta fx ++ -- [ usF ; meta n ] | << meta fep ) imply
  meta fy <<== meta fx end line

line ell f because 1rule mp modus ponens ell e modus ponens ell b indeed
for all meta n indeed ( 00 << meta fep imply meta m <= meta n imply 
  |r meta fx ++ -- [ usF ; meta n ] | << meta fep ) imply
meta fy <<== meta fx end line

because pred lemma exist mp modus ponens ell f modus ponens ell d indeed
  meta fy <<== meta fx end line
line ell big a end block

any term meta m comma meta n comma meta fep comma meta fx comma meta fy end line

line ell a because 1rule deduction modus ponens ell big a indeed
  limit( meta fx , usF ) imply meta fy in0 setOfFxs imply meta fy <<== meta fx end line

line ell b premise limit( meta fx , usF ) end line
line ell c because 1rule mp modus ponens ell a modus ponens ell b indeed
  meta fy in0 setOfFxs imply meta fy <<== meta fx end line

because 1rule toUpperBound modus ponens ell c indeed upperBound( meta fx , setOfFxs ) qed
end math ]"

-------(26.10.06)


"[  math in theory system Q lemma lemma (-1)*(-1)+(-1)*1=0(R) says
(--01) ** (--01) ++ (--01) ** 01 == 00
end lemma end math ]"

"[ math system Q proof of lemma (-1)*(-1)+(-1)*1=0(R) reads
line ell a because lemma distributionOut(R) indeed 
  (--01) ** (--01) ++ (--01) ** 01 == (--01) ** ( (--01) ++ 01 ) end line
line ell b because lemma negative(R)  indeed 01 ++ (--01) == 00  end line
line ell c because lemma plusCommutativity(R) indeed (--01) ++ 01 == 01 ++ (--01) end line
line ell d because lemma ==Transitivity modus ponens ell c modus ponens ell b indeed 
  (--01) ++ 01 == 00 end line
line ell e because lemma eqMultiplicationLeft(R) modus ponens ell d indeed 
  (--01) ** ( (--01) ++ 01 ) == (--01) ** 00 end line

line ell f because lemma x*0=0(R) indeed (--01) ** 00 == 00 end line

because lemma ==Transitivity4 modus ponens ell a modus ponens ell e modus ponens ell f indeed  
(--01) ** (--01) ++ (--01) ** 01 == 00 qed
end math ]"

"[  math in theory system Q lemma lemma (-1)*(-1)=1(R) says
(--01) ** (--01) == 01 
end lemma end math ]"

"[ math system Q proof of lemma (-1)*(-1)=1(R) reads
line ell a because lemma x=x+(y-y)(R) indeed
  (--01) ** (--01) == (--01) ** (--01) ++ ( 01 ++ (--01) ) end line

line ell b because lemma times1(R) indeed (--01) ** 01 == (--01) end line
line ell c because lemma ==Symmetry modus ponens ell b indeed (--01) == (--01) ** 01 end line
line ell d because lemma eqAdditionLeft(R) modus ponens ell c indeed 
  01 ++ (--01) == 01 ++ (--01) ** 01 end line
line ell e because lemma eqAdditionLeft(R) modus ponens ell d indeed 
  (--01) ** (--01) ++ ( 01 ++ (--01) ) == 
  (--01) ** (--01) ++ ( 01 ++ (--01) ** 01 ) end line

line ell f because lemma plusCommutativity(R) indeed 01 ++ (--01) ** 01 == (--01) ** 01 ++ 01 end line 
line ell g because lemma eqAdditionLeft(R) modus ponens ell f indeed 
  (--01) ** (--01) ++ ( 01 ++ (--01) ** 01 ) == 
  (--01) ** (--01) ++ ( (--01) ** 01 ++ 01 ) end line

line ell big a because lemma plusAssociativity(R) indeed
  (--01) ** (--01) ++ (--01) ** 01 ++ 01 == (--01) ** (--01) ++ ( (--01) ** 01 ++ 01 )  end line
line ell h because lemma ==Symmetry modus ponens ell big a indeed
  (--01) ** (--01) ++ ( (--01) ** 01 ++ 01 ) == (--01) ** (--01) ++ (--01) ** 01 ++ 01 end line


line ell i because lemma (-1)*(-1)+(-1)*1=0(R) indeed
  (--01) ** (--01) ++ (--01) ** 01 == 00 end line
line ell j because 1rule adhoc eqAddition(R) modus ponens ell i indeed
  (--01) ** (--01) ++ (--01) ** 01 ++ 01 == 00 ++ 01 end line

line ell k because lemma plus0Left(R) indeed 00 ++ 01 == 01 end line

line ell l because lemma ==Transitivity5 
modus ponens ell a modus ponens ell e modus ponens ell g modus ponens ell h indeed 
  (--01) ** (--01) == (--01) ** (--01) ++ (--01) ** 01 ++ 01 end line
because lemma ==Transitivity4 modus ponens ell l modus ponens ell j modus ponens ell k indeed  
  (--01) ** (--01) == 01 qed
end math ]"


"[  math in theory system Q lemma lemma 0<1Helper(R) says
01 <<== 00 imply 00 <<== 01 
end lemma end math ]"

"[ math system Q proof of lemma 0<1Helper(R) reads
block
line ell a premise 01 <<== 00 end line
line ell b because lemma leqAddition(R) modus ponens ell a indeed 01 ++ (--01) <<== 00 ++ (--01) end line
line ell c because lemma negative(R) indeed 01 ++ (--01) == 00 end line 
line ell d because lemma subLeqLeft(R) modus ponens ell c modus ponens ell b indeed 
  00 <<== 00 ++ (--01) end line
line ell e because lemma plus0Left(R) indeed 00 ++ (--01) == (--01) end line
line ell f because lemma subLeqRight(R) modus ponens ell e modus ponens ell d indeed 00 <<== (--01) end line
line ell g because lemma leqMultiplication(R) modus ponens ell f modus ponens ell f indeed 
  00 ** (--01) <<== (--01) ** (--01) end line
line ell h because lemma x*0=0(R) indeed (--01) ** 00 == 00 end line
line ell i because lemma timesCommutativity(R) indeed 00 ** (--01) == (--01) ** 00 end line
line ell j because lemma ==Transitivity modus ponens ell i modus ponens ell h indeed 
  00 ** (--01) == 00 end line
line ell k because lemma subLeqLeft(R) modus ponens ell j modus ponens ell g indeed 
  00 <<== (--01) ** (--01) end line
line ell l because lemma (-1)*(-1)=1(R) indeed (--01) ** (--01) == 01 end line
because lemma subLeqRight(R) modus ponens ell l modus ponens ell k indeed 00 <<== 01 end line
line ell m end block

because 1rule deduction modus ponens ell m indeed 01 <<== 00 imply 00 <<== 01 qed
end math ]"

"[  math in theory system Q lemma lemma 0<1(R) says
00 << 01 end lemma end math ]"

"[ math system Q proof of lemma 0<1(R) reads
line ell a because lemma leqTotality(R) indeed 00 <<== 01 or0 01 <<== 00 end line
line ell b because prop lemma auto imply indeed 00 <<== 01 imply 00 <<== 01 end line
line ell c because lemma 0<1Helper(R) indeed  01 <<== 00 imply 00 <<== 01 end line

line ell d because prop lemma from disjuncts modus ponens ell a modus ponens ell b modus ponens ell c indeed 00 <<== 01  end line

line ell e because axiom 0not1(R) indeed 00 !!== 01 end line

because lemma leqNeqLess(R) modus ponens ell d modus ponens ell e indeed 00 << 01 qed
end math ]"

-------(26.10.06)


"[  math in theory system Q lemma lemma expZero exact(R) says
for all terms meta fx indeed
meta fx ^ 0 == 01 
end lemma end math ]"

"[ math system Q proof of lemma expZero exact(R) reads
any term meta fx end line
line ell a because lemma eqReflexivity indeed 0 = 0 end line
because 1rule expZero(R) modus ponens ell a indeed meta fx ^ 0 == 01 qed
end math ]"


"[  math in theory system Q lemma lemma positiveBase(R) base says
for all terms meta fx indeed
00 << meta fx ^ 0
end lemma end math ]"

"[ math system Q proof of lemma positiveBase(R) base reads
any term meta fx end line
line ell a because lemma expZero exact(R) indeed meta fx ^ 0 == 01 end line
line ell b because lemma ==Symmetry modus ponens ell a indeed 01 == meta fx ^ 0 end line
line ell c because lemma 0<1(R) indeed 00 << 01 end line

because lemma subLessRight(R) modus ponens ell b modus ponens ell c indeed 00 << meta fx ^ 0 qed
end math ]"


"[ math in theory system Q lemma lemma three2twoFactors(R) says 
for all terms meta fx comma meta fy comma meta fz comma meta fu indeed 
meta fy ** meta fz == meta fu infer 
meta fx ** meta fy ** meta fz == meta fx ** meta fu 
end lemma end math ]"

"[ math system Q proof of lemma three2twoFactors(R) reads
any term meta fx comma meta fy comma meta fz comma meta fu end line
line ell a premise meta fy ** meta fz == meta fu end line
line ell b because lemma eqMultiplicationLeft(R) modus ponens ell a indeed 
  meta fx ** ( meta fy ** meta fz ) == meta fx ** meta fu end line
line ell c because lemma timesAssociativity(R) indeed  
  meta fx ** meta fy ** meta fz == meta fx ** ( meta fy ** meta fz ) end line
because lemma ==Transitivity modus ponens ell c modus ponens ell b indeed  
  meta fx ** meta fy ** meta fz == meta fx ** meta fu qed 
end math ]"



"[  math in theory system Q lemma lemma x=x*y*(1/y)(R) says
for all terms meta fx comma meta fy indeed
meta fy !!== 00 infer meta fx == meta fx ** meta fy ** 01// meta fy
end lemma end math ]"

"[ math system Q proof of lemma x=x*y*(1/y)(R) reads
any term meta fx comma meta fy end line
line ell a premise meta fy !!== 00 end line
line ell b because lemma times1(R) indeed meta fx ** 01 == meta fx end line

line ell c because lemma reciprocal(R) modus ponens ell a indeed meta fy ** 01// meta fy == 01 end line
line ell d because lemma three2twoFactors(R) modus ponens ell c indeed
  meta fx ** meta fy ** 01// meta fy == meta fx ** 01 end line

line ell e because lemma ==Transitivity modus ponens ell d modus ponens ell b indeed
  meta fx ** meta fy ** 01// meta fy == meta fx end line
because lemma ==Symmetry modus ponens ell e indeed meta fx == meta fx ** meta fy ** 01// meta fy qed
end math ]"


"[  math in theory system Q lemma lemma neqMultiplication(R) says
for all terms meta fx comma meta fy comma meta fz indeed
meta fz !!== 00 infer meta fx !!== meta fy infer meta fx ** meta fz !!== meta fy ** meta fz
end lemma end math ]"

"[ math system Q proof of lemma neqMultiplication(R) reads
block any term meta fx comma meta fy comma meta fz end line
line ell a premise meta fz !!== 00 end line
line ell b premise meta fx !!== meta fy end line

line ell c premise meta fx ** meta fz == meta fy ** meta fz end line

line ell d because lemma x=x*y*(1/y)(R) modus ponens ell a indeed
  meta fx == meta fx ** meta fz ** 01// meta fz end line

line ell e because lemma eqMultiplication(R) modus ponens ell c indeed
  meta fx ** meta fz ** 01// meta fz == meta fy ** meta fz ** 01// meta fz end line

line ell f because lemma x=x*y*(1/y)(R) modus ponens ell a indeed
  meta fy == meta fy ** meta fz ** 01// meta fz end line
line ell g because lemma ==Symmetry modus ponens ell f indeed
  meta fy ** meta fz ** 01// meta fz == meta fy end line

line ell h because lemma ==Transitivity4 modus ponens ell d modus ponens ell e modus ponens ell g indeed
  meta fx == meta fy end line

because prop lemma from contradiction modus ponens ell h modus ponens ell b indeed
  meta fx ** meta fz !!== meta fy ** meta fz end line
line ell i end block

any term meta fx comma meta fy comma meta fz end line
line ell j because 1rule deduction modus ponens ell i indeed
  meta fz !!== 00 imply meta fx !!== meta fy imply 
  meta fx ** meta fz == meta fy ** meta fz imply meta fx ** meta fz !!== meta fy ** meta fz end line

line ell k premise meta fz !!== 00 end line
line ell l premise meta fx !!== meta fy end line

line ell m  because prop lemma mp2 modus ponens ell j modus ponens ell k modus ponens ell l indeed
  meta fx ** meta fz == meta fy ** meta fz imply meta fx ** meta fz !!== meta fy ** meta fz end line

because prop lemma imply negation modus ponens ell m indeed 
  meta fx ** meta fz !!== meta fy ** meta fz qed
end math ]"


--------(26.10.06)

 



"[  math in theory system Q lemma lemma lessTransitivity(R) says
for all terms meta fx comma meta fy comma meta fz indeed
meta fx << meta fy infer meta fy << meta fz infer meta fx << meta fz 
end lemma end math ]"

"[ math system Q proof of lemma lessTransitivity(R) reads
any term meta fx comma meta fy comma meta fz end line
line ell a premise meta fx << meta fy end line
line ell b premise meta fy << meta fz end line
line ell c because lemma lessLeq(R) modus ponens ell a indeed meta fx <<== meta fy end line

because lemma leqLessTransitivity(R) modus ponens ell c modus ponens ell b indeed meta fx << meta fz qed
end math ]"

"[  math in theory system Q lemma lemma 0<2(R) says
00 << 02 end lemma end math ]"

"[ math system Q proof of lemma 0<2(R) reads
line ell a because lemma 0<1(R) indeed 00 << 01 end line

line ell c because lemma lessAddition(R) modus ponens ell a indeed
  00 ++ 01 << 01 ++ 01 end line
line ell d because lemma plus0Left(R) indeed 00 ++ 01 == 01 end line
line ell e because lemma subLessLeft(R) modus ponens ell d modus ponens ell c indeed
  01 << 01 ++ 01 end line

because lemma lessTransitivity(R) modus ponens ell a modus ponens ell e indeed  
  00 << 02 qed
end math ]"


"[  math in theory system Q lemma lemma sameExp(R) base says
for all terms meta n comma meta fx indeed
for all meta n indeed ( 0 = meta n imply meta fx ^ 0 == meta fx ^ meta n )
end lemma end math ]"


"[ math system Q proof of lemma sameExp(R) base reads
block any term meta n comma meta fx end line
line ell a premise 0 = meta n end line
line ell b because lemma expZero exact(R) indeed meta fx ^ 0 == 01 end line

line ell c because lemma eqSymmetry modus ponens ell a indeed meta n = 0 end line
line ell d because 1rule expZero(R) modus ponens ell c indeed meta fx ^ meta n == 01 end line
line ell e because lemma ==Symmetry modus ponens ell d indeed 01 == meta fx ^ meta n end line

because lemma ==Transitivity modus ponens ell b modus ponens ell e indeed
  meta fx ^ 0 == meta fx ^ meta n end line
line ell big a end block

any term meta n comma meta fx end line

line ell a because 1rule deduction modus ponens ell big a indeed
  0 = meta n imply meta fx ^ 0 == meta fx ^ meta n end line
because 1rule gen modus ponens ell a indeed 
  for all meta n indeed ( 0 = meta n imply meta fx ^ 0 == meta fx ^ meta n ) qed
end math ]"

"[  math in theory system Q lemma lemma sameExp(R) indu says
for all terms meta m comma meta n comma meta fx indeed
  for all meta n indeed ( meta m = meta n imply meta fx ^ meta m == meta fx ^ meta n ) imply
  for all meta n indeed ( meta m + 1 = meta n imply meta fx ^ ( meta m + 1 ) == meta fx ^ meta n )
end lemma end math ]"


"[ math system Q proof of lemma sameExp(R) indu reads

block any term meta m comma meta n comma meta fx end line
block any term meta m comma meta n comma meta fx end line
line ell a premise for all meta n indeed 
  ( meta m = meta n imply meta fx ^ meta m == meta fx ^ meta n ) end line
line ell b premise meta m + 1 = meta n end line

line ell c because lemma +1IsPositive(N) indeed 0 < meta m + 1 end line
line ell f because 1rule expPositive(R) modus ponens ell c indeed
  meta fx ^ ( meta m + 1 ) == 
  meta fx ** meta fx ^ ( meta m + 1 - 1 ) end line

line ell g because lemma x=x+y-y indeed meta m = meta m + 1 - 1 end line
line ell h because lemma a4 at meta m + 1 - 1 modus ponens ell a indeed
  meta m = meta m + 1 - 1 imply 
  meta fx ^ meta m == meta fx ^ ( meta m + 1 - 1 ) end line
line ell i because 1rule mp modus ponens ell h modus ponens ell g indeed
  meta fx ^ meta m == meta fx ^ ( meta m + 1 - 1 ) end line
line ell j because lemma ==Symmetry modus ponens ell i indeed
  meta fx ^ ( meta m + 1 - 1 ) == meta fx ^ meta m end line
line ell k because lemma eqMultiplicationLeft(R) modus ponens ell j indeed
  meta fx ** meta fx ^ ( meta m + 1 - 1 ) == meta fx ** meta fx ^ meta m end line

line ell l because lemma a4 at meta n - 1 modus ponens ell a indeed
  meta m = meta n - 1 imply meta fx ^ meta m == meta fx ^ ( meta n - 1 ) end line
line ell m because lemma positiveToRight(Eq) modus ponens ell b indeed
  meta m = meta n - 1 end line
line ell o because 1rule mp modus ponens ell l modus ponens ell m indeed
  meta fx ^ meta m == meta fx ^ ( meta n - 1 ) end line
line ell p because lemma eqMultiplicationLeft(R) modus ponens ell o indeed
  meta fx ** meta fx ^ meta m == meta fx ** meta fx ^ ( meta n - 1 ) end line


line ell q because lemma subLessRight modus ponens ell b modus ponens ell c indeed 0 < meta n end line
line ell r because 1rule expPositive(R) modus ponens ell q indeed 
  meta fx ^ meta n == meta fx ** meta fx ^ ( meta n - 1 ) end line
line ell s because lemma ==Symmetry modus ponens ell r indeed
  meta fx ** meta fx ^ ( meta n - 1 ) == meta fx ^ meta n end line

because lemma ==Transitivity5 modus ponens ell f modus ponens ell k modus ponens ell p modus ponens ell s
  indeed meta fx ^ ( meta m + 1 ) == meta fx ^ meta n end line
line ell big a end block

line ell a because 1rule deduction modus ponens ell big a indeed 
for all meta n indeed 
  ( meta m = meta n imply meta fx ^ meta m == meta fx ^ meta n ) imply
meta m + 1 = meta n imply
meta fx ^ ( meta m + 1 ) == meta fx ^ meta n end line

line ell b premise for all meta n indeed 
  ( meta m = meta n imply meta fx ^ meta m == meta fx ^ meta n ) end line

line ell c because 1rule mp modus ponens ell a modus ponens ell b indeed
  meta m + 1 = meta n imply meta fx ^ ( meta m + 1 ) == meta fx ^ meta n end line

because 1rule gen modus ponens ell c indeed for all meta n indeed ( meta m + 1 = meta n imply
  meta fx ^ ( meta m + 1 ) == meta fx ^ meta n ) end line
line ell big b end block

any term meta m comma meta n comma meta fx end line 

because 1rule deduction modus ponens ell big b indeed  
for all meta n indeed 
  ( meta m = meta n imply meta fx ^ meta m == meta fx ^ meta n ) imply
for all meta n indeed ( meta m + 1 = meta n imply
  meta fx ^ ( meta m + 1 ) == meta fx ^ meta n ) qed
end math ]"

"[  math in theory system Q lemma lemma sameExp(R) says
for all terms meta m comma meta n comma meta fx indeed
  meta m = meta n infer meta fx ^ meta m == meta fx ^ meta n
end lemma end math ]"

"[ math system Q proof of lemma sameExp(R) reads


any term meta m comma meta n comma meta fx end line

line ell big a premise meta m = meta n end line
line ell a because lemma sameExp(R) base indeed
  for all meta n indeed ( 0 = meta n imply meta fx ^ 0 == meta fx ^ meta n )
  end line
line ell b because lemma sameExp(R) indu indeed for all meta n indeed 
  ( meta m = meta n imply meta fx ^ meta m == meta fx ^ meta n ) imply
  for all meta n indeed ( meta m + 1 = meta n imply 
  meta fx ^ ( meta m + 1 ) == meta fx ^ meta n ) end line
line ell c because lemma induction modus ponens ell a modus ponens ell b indeed
  for all meta n indeed ( meta m = meta n imply meta fx ^ meta m == meta fx ^ meta n )
  end line
line ell d because lemma a4 at meta n modus ponens ell c indeed 
  meta m = meta n imply meta fx ^ meta m == meta fx ^ meta n end line

because 1rule mp modus ponens ell d modus ponens ell big a indeed meta fx ^ meta m == meta fx ^ meta n qed
end math ]"

-------(26.10.06)


"[  math in theory system Q lemma lemma subNeqLeft(R) says
for all terms meta fx comma meta fy comma meta fz indeed
meta fx == meta fy infer meta fx !!== meta fz infer meta fy !!== meta fz 
end lemma end math ]"

"[ math system Q proof of lemma subNeqLeft(R) reads
block any term meta fx comma meta fy comma meta fz end line
line ell a premise meta fx == meta fy end line
line ell b premise meta fx !!== meta fz end line
line ell c premise meta fy == meta fz end line
line ell d because lemma ==Transitivity modus ponens ell a modus ponens ell c indeed 
  meta fx == meta fz end line
because prop lemma from contradiction modus ponens ell d modus ponens ell b indeed
  meta fy !!== meta fz end line
line ell big a end block

any term meta fx comma meta fy comma meta fz end line

line ell a because 1rule deduction modus ponens ell big a indeed
  meta fx == meta fy imply meta fx !!== meta fz imply meta fy == meta fz imply meta fy !!== meta fz end line

line ell b premise meta fx == meta fy end line
line ell c premise meta fx !!== meta fz end line

line ell d because prop lemma mp2 modus ponens ell a modus ponens ell b modus ponens ell c indeed
  meta fy == meta fz imply meta fy !!== meta fz end line

because prop lemma imply negation modus ponens ell d indeed meta fy !!== meta fz qed
end math ]"


"[  math in theory system Q lemma lemma subNeqRight(R) says
for all terms meta fx comma meta fy comma meta fz indeed
meta fx == meta fy infer meta fz !!== meta fx infer meta fz !!== meta fy
end lemma end math ]"

"[ math system Q proof of lemma subNeqRight(R) reads
any term meta fx comma meta fy comma meta fz end line
line ell a premise meta fx == meta fy end line
line ell b premise meta fz !!== meta fx end line
line ell c because lemma !!==Symmetry modus ponens ell b indeed meta fx !!== meta fz end line
line ell d because lemma subNeqLeft(R) modus ponens ell a modus ponens ell c indeed
  meta fy !!== meta fz end line
because lemma !!==Symmetry modus ponens ell d indeed meta fz !!== meta fy qed
end math ]"

"[  math in theory system Q lemma lemma nonzeroFactors(R) says
for all terms meta fx comma meta fy indeed
meta fx !!== 00 infer meta fy !!== 00 infer meta fx ** meta fy !!== 00 
end lemma end math ]"

"[ math system Q proof of lemma nonzeroFactors(R) reads
any term meta fx comma meta fy end line
line ell a premise meta fx !!== 00 end line
line ell b premise meta fy !!== 00 end line
line ell c because lemma neqMultiplication(R) modus ponens ell b modus ponens ell a indeed
  meta fx ** meta fy !!== 00 ** meta fy end line

line ell d because lemma timesCommutativity(R) indeed 00 ** meta fy == meta fy ** 00 end line
line ell e because lemma x*0=0(R) indeed meta fy ** 00 == 00 end line ""; XX muligt lemma her
line ell f because lemma ==Transitivity modus ponens ell d modus ponens ell e indeed 
  00 ** meta fy == 00 end line

because lemma subNeqRight(R) modus ponens ell f modus ponens ell c indeed 
 meta fx ** meta fy !!== 00 qed
end math ]"

"[  math in theory system Q lemma lemma nonnegativeFactors(R) says
for all terms meta fx comma meta fy indeed
00 <<== meta fx infer 00 <<== meta fy infer 00 <<== meta fx ** meta fy 
end lemma end math ]"

"[ math system Q proof of lemma nonnegativeFactors(R) reads
any term meta fx comma meta fy end line
line ell a premise 00 <<== meta fx end line
line ell b premise 00 <<== meta fy end line
line ell c because lemma leqMultiplication(R) modus ponens ell b modus ponens ell a indeed
  00 ** meta fy <<== meta fx ** meta fy end line

line ell d because lemma timesCommutativity(R) indeed 00 ** meta fy == meta fy ** 00 end line
line ell e because lemma x*0=0(R) indeed meta fy ** 00 == 00 end line
line ell f because lemma ==Transitivity modus ponens ell d modus ponens ell e indeed 
  00 ** meta fy == 00 end line
because lemma subLeqLeft(R) modus ponens ell f modus ponens ell c indeed  00 <<== meta fx ** meta fy qed
end math ]"


"[  math in theory system Q lemma lemma positiveFactors(R) says
for all terms meta fx comma meta fy indeed
00 << meta fx infer 00 << meta fy infer 00 << meta fx ** meta fy
end lemma end math ]"

"[ math system Q proof of lemma positiveFactors(R) reads
any term meta fx comma meta fy end line
line ell a premise 00 << meta fx end line
line ell b premise 00 << meta fy end line

line ell d because lemma lessLeq(R) modus ponens ell a indeed 00 <<== meta fx end line
line ell e because lemma lessNeq(R) modus ponens ell a indeed 00 !!== meta fx end line
line ell f because lemma !!==Symmetry modus ponens ell e indeed meta fx !!== 00 end line

line ell h because lemma lessLeq(R) modus ponens ell b indeed 00 <<== meta fy end line
line ell i because lemma lessNeq(R) modus ponens ell b indeed 00 !!== meta fy end line
line ell j because lemma !!==Symmetry modus ponens ell i indeed meta fy !!== 00 end line

line ell k because lemma nonnegativeFactors(R) modus ponens ell d modus ponens ell h indeed
  00 <<== meta fx ** meta fy end line
line ell l because lemma nonzeroFactors(R) modus ponens ell f modus ponens ell j indeed
  meta fx ** meta fy !!== 00 end line
line ell m because lemma !!==Symmetry modus ponens ell l indeed 00 !!== meta fx ** meta fy end line

because lemma leqNeqLess(R) modus ponens ell k modus ponens ell m indeed 00 << meta fx ** meta fy qed
end math ]"

--------(26.10.06)

"[  math in theory system Q lemma lemma lessDivision(R) says
for all terms meta fx comma meta fy comma meta fz indeed
00 <<== meta fz infer meta fx ** meta fz << meta fy ** meta fz infer meta fx << meta fy 
end lemma end math ]"

"[ math system Q proof of lemma lessDivision(R) reads
block any term meta fx comma meta fy comma meta fz end line

line ell a premise 00 <<== meta fz end line
line ell b premise meta fx ** meta fz << meta fy ** meta fz end line
line ell c premise meta fy <<== meta fx end line

line ell d because lemma leqMultiplication(R) modus ponens ell a modus ponens ell c indeed
  meta fy ** meta fz <<== meta fx ** meta fz end line
  
line ell e because lemma fromLess(R) modus ponens ell b indeed 
  not0 meta fy ** meta fz <<== meta fx ** meta fz end line

because prop lemma from contradiction modus ponens ell d modus ponens ell e indeed
  not0 meta fy <<== meta fx end line
line ell big a end block

any term meta fx comma meta fy comma meta fz end line

line ell a because 1rule deduction modus ponens ell big a indeed
  00 <<== meta fz imply meta fx ** meta fz << meta fy ** meta fz imply 
  meta fy <<== meta fx imply not0 meta fy <<== meta fx end line

line ell b premise 00 <<== meta fz end line
line ell c premise meta fx ** meta fz << meta fy ** meta fz end line

line ell d because prop lemma mp2 modus ponens ell a modus ponens ell b modus ponens ell c indeed
  meta fy <<== meta fx imply not0 meta fy <<== meta fx end line

line ell e because prop lemma imply negation modus ponens ell d indeed not0 meta fy <<== meta fx end line

because lemma toLess(R) modus ponens ell e indeed meta fx << meta fy qed
end math ]"

"[  math in theory system Q lemma lemma 0<1/2(R) says
00 << 01//02 end lemma end math ]"

"[ math system Q proof of lemma 0<1/2(R) reads
line ell a because lemma 0<2(R) indeed 00 << 02 end line
line ell b because lemma lessLeq(R) modus ponens ell a indeed 00 <<== 02 end line
line ell c because lemma lessNeq(R) modus ponens ell a indeed 00 !!== 02 end line
line ell d because lemma !!==Symmetry modus ponens ell c indeed 02 !!== 00 end line


line ell e because lemma 0<1(R) indeed 00 << 01 end line
line ell f because lemma x*0=0(R) indeed 02 ** 00 == 00 end line
line ell g because lemma ==Symmetry modus ponens ell f indeed 00 == 02 ** 00 end line
line ell h because lemma timesCommutativity(R) indeed 02 ** 00 == 00 ** 02 end line
line ell i because lemma ==Transitivity modus ponens ell g modus ponens ell h indeed
  00 == 00 ** 02 end line

line ell j because lemma subLessLeft(R) modus ponens ell i modus ponens ell e indeed 
  00 ** 02 << 01 end line
line ell k because lemma reciprocal(R) modus ponens ell d indeed 02 ** 01//02 == 01 end line

line ell l because lemma ==Symmetry modus ponens ell k indeed 01 == 02 ** 01//02 end line
line ell m because lemma timesCommutativity(R) indeed 02 ** 01//02 == 01//02 ** 02 end line
line ell n because lemma ==Transitivity modus ponens ell l modus ponens ell m indeed
  01 == 01//02 ** 02 end line

line ell o because lemma subLessRight(R) modus ponens ell n modus ponens ell j indeed
  00 ** 02 << 01//02 ** 02 end line
because lemma lessDivision(R) modus ponens ell b modus ponens ell o indeed 00 << 01//02 qed
end math ]"


"[  math in theory system Q lemma lemma positiveToRight(Eq)(1 term)(R) says
for all terms meta fx comma meta fy indeed
meta fx == meta fy infer 00 == meta fy ++ -- meta fx
end lemma end math ]"

"[ math system Q proof of lemma positiveToRight(Eq)(1 term)(R) reads
any term meta fx comma meta fy end line
line ell a premise meta fx == meta fy end line

line ell c because lemma plusCommutativity(R) indeed 00 ++ meta fx == meta fx ++ 00 end line
line ell b because lemma plus0(R) indeed meta fx ++ 00 == meta fx end line

line ell d because lemma ==Transitivity4 modus ponens ell c modus ponens ell b modus ponens ell a indeed
  00 ++ meta fx == meta fy end line

because lemma positiveToRight(Eq)(R) modus ponens ell d indeed 00 == meta fy ++ -- meta fx qed
end math ]"

"[  math in theory system Q lemma lemma exp(+1)(R) says
for all terms meta m comma meta fx indeed
meta fx ^ ( meta m + 1 ) == meta fx ** meta fx ^ meta m 
end lemma end math ]"

"[ math system Q proof of lemma exp(+1)(R) reads
any term meta m comma meta fx end line
line ell a because lemma +1IsPositive(N) indeed 0 < meta m + 1 end line
line ell b because 1rule expPositive(R) modus ponens ell a indeed
  meta fx ^ ( meta m + 1 ) == 
  meta fx ** meta fx ^ ( meta m + 1 - 1 ) end line

line ell c because lemma x=x+y-y indeed meta m = meta m + 1 - 1 end line
line ell d because lemma sameExp(R) modus ponens ell c indeed
  meta fx ^ meta m == meta fx ^ ( meta m + 1 - 1 ) end line
line ell e because lemma eqMultiplicationLeft(R) modus ponens ell d indeed
  meta fx ** meta fx ^ meta m == meta fx ** meta fx ^ ( meta m + 1 - 1 ) end line
line ell f because lemma ==Symmetry modus ponens ell e indeed
   meta fx ** meta fx ^ ( meta m + 1 - 1 ) == meta fx ** meta fx ^ meta m end line

because lemma ==Transitivity modus ponens ell b modus ponens ell f indeed 
  meta fx ^ ( meta m + 1 ) == meta fx ** meta fx ^ meta m qed
end math ]"

"[  math in theory system Q lemma lemma positiveBase(R) indu says
for all terms meta m comma meta fx indeed
00 << meta fx infer 00 << meta fx ^ meta m imply 00 << meta fx ^ ( meta m + 1 )
end lemma end math ]"

"[ math system Q proof of lemma positiveBase(R) indu reads
block any term meta m comma meta fx end line
line ell a premise 00 << meta fx end line
line ell b premise 00 << meta fx ^ meta m end line
line ell c because lemma exp(+1)(R) indeed 
  meta fx ^ ( meta m + 1 ) == meta fx ** meta fx ^ meta m end line
line ell d because lemma ==Symmetry modus ponens ell c indeed
  meta fx ** meta fx ^ meta m ==  meta fx ^ ( meta m + 1 ) end line

line ell e because lemma positiveFactors(R) modus ponens ell a modus ponens ell b indeed
  00 << meta fx ** meta fx ^ meta m end line
because lemma subLessRight(R) modus ponens ell d modus ponens ell e indeed
  00 << meta fx ^ ( meta m + 1 ) end line
line ell big a end block

any term meta m comma meta fx end line

line ell a because 1rule deduction modus ponens ell big a indeed
  00 << meta fx imply 00 << meta fx ^ meta m imply 00 << meta fx ^ ( meta m + 1 ) end line
line ell b premise 00 << meta fx end line

because 1rule mp modus ponens ell a modus ponens ell b indeed 
 00 << meta fx ^ meta m imply 00 << meta fx ^ ( meta m + 1 ) qed
end math ]"

"[  math in theory system Q lemma lemma positiveBase(R) says
for all terms meta m comma meta fx indeed
00 << meta fx infer 00 << meta fx ^ meta m 
end lemma end math ]"

"[ math system Q proof of lemma positiveBase(R) reads
any term meta m comma meta fx end line
line ell a premise 00 << meta fx end line
line ell b because lemma positiveBase(R) base indeed 00 << meta fx ^ 0 end line
line ell c because lemma positiveBase(R) indu modus ponens ell a indeed
  00 << meta fx ^ meta m imply 00 << meta fx ^ ( meta m + 1 ) end line

because lemma induction modus ponens ell b modus ponens ell c indeed 00 << meta fx ^ meta m qed
end math ]"

-------(26.10.06)

"[  math in theory system Q lemma lemma -x*y=-(x*y)(R) says
for all terms meta fx comma meta fy indeed
-- meta fx ** meta fy == -- ( meta fx ** meta fy )
end lemma end math ]"

"[ math system Q proof of lemma -x*y=-(x*y)(R) reads
any term meta fx comma meta fy end line
line ell a because lemma times(-1)Left(R) indeed (--01) ** meta fx == -- meta fx end line
line ell b because lemma eqMultiplication(R) modus ponens ell a indeed 
  (--01) ** meta fx ** meta fy == -- meta fx ** meta fy end line
line ell c because lemma ==Symmetry modus ponens ell b indeed
  -- meta fx ** meta fy == (--01) ** meta fx ** meta fy end line

line ell d because lemma timesAssociativity(R) indeed 
  (--01) ** meta fx ** meta fy == (--01) ** ( meta fx ** meta fy ) end line

line ell e because lemma times(-1)Left(R) indeed 
  (--01) ** ( meta fx ** meta fy ) == -- ( meta fx ** meta fy ) end line 

because lemma ==Transitivity4 modus ponens ell c modus ponens ell d modus ponens ell e indeed 
  -- meta fx ** meta fy == -- ( meta fx ** meta fy ) qed
end math ]"


"[  math in theory system Q lemma lemma positiveToLeft(Eq)(R) says
for all terms meta fx comma meta fy comma meta fz indeed
meta fx == meta fy ++ meta fz infer meta fx ++ -- meta fz == meta fy
end lemma end math ]"

"[ math system Q proof of lemma positiveToLeft(Eq)(R) reads
any term meta fx comma meta fy comma meta fz end line
line ell a premise meta fx == meta fy ++ meta fz end line
line ell b because 1rule adhoc eqAddition(R) modus ponens ell a indeed 
  meta fx ++ -- meta fz == meta fy ++ meta fz ++ -- meta fz end line

line ell c because lemma x=x+y-y(R) indeed
  meta fy == meta fy ++ meta fz ++ -- meta fz end line
line ell d because lemma ==Symmetry modus ponens ell c indeed 
  meta fy ++ meta fz ++ -- meta fz == meta fy end line

because lemma ==Transitivity modus ponens ell b modus ponens ell d indeed
  meta fx ++ -- meta fz == meta fy qed
end math ]"

"[  math in theory system Q lemma lemma times1Left(R) says
for all terms meta fx indeed
01 ** meta fx == meta fx 
end lemma end math ]"

"[ math system Q proof of lemma times1Left(R) reads
any term meta fx end line
line ell a because lemma times1(R) indeed meta fx ** 01 == meta fx end line
line ell b because lemma timesCommutativity(R) indeed 01 ** meta fx == meta fx ** 01 end line
because lemma ==Transitivity modus ponens ell b  modus ponens ell a indeed 01 ** meta fx == meta fx qed
end math ]"

"[  math in theory system Q lemma lemma ==Transitivity6 says
for all terms meta fx comma meta fy comma meta fz comma meta fu comma meta fv comma meta fw indeed
meta fx == meta fy infer meta fy == meta fz infer meta fz == meta fu infer
meta fu == meta fv infer meta fv == meta fw infer meta fx == meta fw
end lemma end math ]"

"[ math system Q proof of lemma ==Transitivity6 reads
any term meta fx comma meta fy comma meta fz comma meta fu comma meta fv comma meta fw end line
line ell a premise meta fx == meta fy end line
line ell b premise meta fy == meta fz end line
line ell c premise meta fz == meta fu end line
line ell d premise meta fu == meta fv end line
line ell e premise meta fv == meta fw end line
line ell f because lemma ==Transitivity5 modus ponens ell a modus ponens ell b modus ponens ell c 
  modus ponens ell d indeed meta fx == meta fv end line
because lemma ==Transitivity modus ponens ell f modus ponens ell e indeed 
meta fx == meta fw qed
end math ]"


"[  math in theory system Q lemma lemma x+x=2*x(R) says
for all terms meta fx indeed
meta fx ++ meta fx == 02 ** meta fx 
end lemma end math ]"

"[ math system Q proof of lemma x+x=2*x(R) reads
any term meta fx end line

line ell a because lemma times1(R)  indeed meta fx ** 01 == meta fx end line
line ell b because lemma ==Symmetry indeed meta fx == meta fx ** 01 end line
line ell c because lemma eqAdditionLeft(R) modus ponens ell b indeed 
  meta fx ++ meta fx == meta fx ++ meta fx ** 01 end line
line ell d because 1rule adhoc eqAddition(R) modus ponens ell b indeed 
  meta fx ++ meta fx ** 01 == meta fx ** 01 ++ meta fx ** 01 end line 
line ell e because lemma ==Transitivity modus ponens ell c modus ponens ell d indeed 
  meta fx ++ meta fx == meta fx ** 01 ++ meta fx ** 01 end line

line ell f because lemma distributionOut(R) indeed 
  meta fx ** 01 ++ meta fx ** 01 == meta fx ** ( 01 ++ 01 ) end line 
line ell g because 1rule repetition modus ponens ell f indeed ""; XXREP 02 
  meta fx ** 01 ++ meta fx ** 01 == meta fx ** 02 end line

line ell h because lemma timesCommutativity(R) indeed meta fx ** 02 == 02 ** meta fx end line

because lemma ==Transitivity4 modus ponens ell e modus ponens ell g modus ponens ell h indeed  
  meta fx ++ meta fx == 02 ** meta fx qed
end math ]"




"[  math in theory system Q lemma lemma (1/2)x+(1/2)x=x(R) says
for all terms meta fx indeed
01//02 ** meta fx ++ 01//02 ** meta fx == meta fx
end lemma end math ]"

"[ math system Q proof of lemma (1/2)x+(1/2)x=x(R) reads
any term meta fx end line
line ell a because lemma 0<2(R) indeed 00 << 02 end line
line ell b because lemma lessNeq(R) modus ponens ell a indeed  00 !!== 02 end line
line ell c because lemma !!==Symmetry modus ponens ell b indeed 02 !!== 00 end line
line ell d because lemma x+x=2*x(R) indeed
  01//02 ** meta fx ++ 01//02 ** meta fx == 02 ** ( 01//02 ** meta fx ) end line

line ell e because lemma timesAssociativity(R) indeed
  02 ** 01//02 ** meta fx == 02 ** ( 01//02 ** meta fx ) end line
line ell big a because lemma ==Symmetry modus ponens ell e indeed
  02 ** ( 01//02 ** meta fx ) == 02 ** 01//02 ** meta fx end line

line ell g because lemma reciprocal(R) modus ponens ell c indeed 02 ** 01//02 == 01 end line
line ell h because lemma eqMultiplication(R) modus ponens ell g indeed 
  02 ** 01//02 ** meta fx == 01 ** meta fx end line

line ell i because lemma times1Left(R) indeed 01 ** meta fx == meta fx end line

because lemma ==Transitivity5 modus ponens ell d modus ponens ell big a 
  modus ponens ell h modus ponens ell i indeed 01//02 ** meta fx ++ 01//02 ** meta fx == meta fx qed
end math ]"



"[  math in theory system Q lemma lemma distributionOut(Minus)(R) says
for all terms meta fx comma meta fy comma meta fz indeed
meta fx ** meta fy ++ -- ( meta fx ** meta fz ) == meta fx ** ( meta fy ++ -- meta fz )
end lemma end math ]"

"[ math system Q proof of lemma distributionOut(Minus)(R) reads

any term meta fx comma meta fy comma meta fz end line
line ell a because lemma times(-1)Left(R) indeed 
 (--01) ** ( meta fx ** meta fz ) == -- ( meta fx ** meta fz ) end line
line ell b because lemma ==Symmetry modus ponens ell a indeed
 -- ( meta fx ** meta fz ) == (--01) ** ( meta fx ** meta fz ) end line

line ell c because lemma timesCommutativity(R) indeed
  (--01) ** ( meta fx ** meta fz ) == meta fx ** meta fz ** (--01) end line

line ell d because lemma timesAssociativity(R) indeed
  meta fx ** meta fz ** (--01) == meta fx ** ( meta fz ** (--01) ) end line

line ell e because lemma times(-1)(R) indeed meta fz ** (--01) ==  -- meta fz end line
line ell f because lemma eqMultiplicationLeft(R) modus ponens ell e indeed
 meta fx ** ( meta fz ** (--01) ) == meta fx ** ( -- meta fz ) end line

line ell g because lemma ==Transitivity5 modus ponens ell b modus ponens ell c modus ponens ell d
  modus ponens ell f indeed -- ( meta fx ** meta fz ) == 
  meta fx ** ( -- meta fz ) end line

line ell h because lemma eqAdditionLeft(R) modus ponens ell g indeed
  meta fx ** meta fy ++ -- ( meta fx ** meta fz ) == 
  meta fx ** meta fy ++ meta fx ** ( -- meta fz ) end line

line ell i because lemma distributionOut(R) indeed 
  meta fx ** meta fy ++ meta fx ** ( -- meta fz ) == 
  meta fx ** ( meta fy ++ -- meta fz ) end line

because lemma ==Transitivity modus ponens ell h modus ponens ell i indeed 
  meta fx ** meta fy ++ -- ( meta fx ** meta fz ) == meta fx ** ( meta fy ++ -- meta fz ) qed
end math ]"


"[ math in theory system Q lemma lemma (1/2)(x+y)-x=(1/2)(y-x)(R) says
for all terms meta fx comma meta fy indeed
01//02 ** ( meta fx ++ meta fy ) ++ -- meta fx == 
01//02 ** ( meta fy ++ -- meta fx )
end lemma end math ]"

"[ math system Q proof of lemma (1/2)(x+y)-x=(1/2)(y-x)(R) reads
any term meta fx comma meta fy end line
line ell a because lemma distribution(R) indeed 
  01//02 ** ( meta fx ++ meta fy ) == 01//02 ** meta fx ++ 01//02 ** meta fy end line
line ell b because 1rule adhoc eqAddition(R) modus ponens ell a indeed
  01//02 ** ( meta fx ++ meta fy ) ++ -- meta fx == 
  01//02 ** meta fx ++ 01//02 ** meta fy ++ -- meta fx end line

line ell c because lemma plusCommutativity(R) indeed 
  01//02 ** meta fx ++ 01//02 ** meta fy == 01//02 ** meta fy ++ 01//02 ** meta fx end line
line ell d because 1rule adhoc eqAddition(R) modus ponens ell c indeed 
  01//02 ** meta fx ++ 01//02 ** meta fy ++ -- meta fx == 
  01//02 ** meta fy ++ 01//02 ** meta fx ++ -- meta fx end line

line ell big a because lemma plusAssociativity(R) indeed
  01//02 ** meta fy ++ 01//02 ** meta fx ++ -- meta fx == 
  01//02 ** meta fy ++ ( 01//02 ** meta fx ++ -- meta fx ) end line

line ell e because lemma (1/2)x+(1/2)x=x(R) indeed 
  01//02 ** meta fx ++ 01//02 ** meta fx == meta fx end line
line ell f because lemma positiveToRight(Eq)(R) modus ponens ell e indeed 
  01//02 ** meta fx == meta fx ++ -- ( 01//02 ** meta fx ) end line
line ell g because lemma eqNegated(R) modus ponens ell f indeed
  -- ( 01//02 ** meta fx ) == -- ( meta fx ++ -- ( 01//02 ** meta fx ) ) end line

"";line ell big b 
"";because lemma -x*y=-(x*y)(R) indeed -- 01//02 ** meta fx == -- ( 01//02 ** meta fx ) end line

line ell h because lemma minusNegated(R) indeed 
  -- ( meta fx ++ -- ( 01//02 ** meta fx ) ) == 01//02 ** meta fx ++ -- meta fx end line

line ell i because lemma ==Transitivity modus ponens ell g modus ponens ell h indeed
  -- ( 01//02 ** meta fx ) == 01//02 ** meta fx ++ -- meta fx end line
line ell j because lemma ==Symmetry modus ponens ell i indeed
  01//02 ** meta fx ++ -- meta fx == -- ( 01//02 ** meta fx ) end line
line ell k because lemma eqAdditionLeft(R) modus ponens ell j indeed
  01//02 ** meta fy ++ ( 01//02 ** meta fx ++ -- meta fx ) == 
  01//02 ** meta fy ++ -- ( 01//02 ** meta fx ) end line

line ell l because lemma distributionOut(Minus)(R) indeed
  01//02 ** meta fy ++ -- ( 01//02 ** meta fx ) == 01//02 ** ( meta fy ++ -- meta fx ) end line

because lemma ==Transitivity6 
  modus ponens ell b modus ponens ell d modus ponens ell big a modus ponens ell k modus ponens ell l indeed 
  01//02 ** ( meta fx ++ meta fy ) ++ -- meta fx == 
  01//02 ** ( meta fy ++ -- meta fx ) qed
end math ]"

"[  math in theory system Q lemma lemma intervalSize(R) base says
[ us ; 0 ] ++ -- [ xs ; 0 ] == 01//02 ^ 0 
end lemma end math ]"

"[ math system Q proof of lemma intervalSize(R) base reads
line ell a because axiom US0 indeed [ us ; 0 ] == [ xs ; 0 ] ++ 01 end line
line ell b because lemma plusCommutativity(R) indeed [ xs ; 0 ] ++ 01 == 01 ++ [ xs ; 0 ] end line
line ell c because lemma ==Transitivity modus ponens ell a modus ponens ell b indeed 
  [ us ; 0 ] == 01 ++ [ xs ; 0 ] end line
line ell d because lemma positiveToLeft(Eq)(R) modus ponens ell c indeed 
  [ us ; 0 ] ++ -- [ xs ; 0 ] == 01 end line

line ell f because lemma expZero exact(R) indeed 01//02 ^ 0 == 01 end line  
line ell h because lemma ==Symmetry modus ponens ell f indeed 01 == 01//02 ^ 0 end line

because lemma ==Transitivity modus ponens ell d modus ponens ell h indeed 
  [ us ; 0 ] ++ -- [ xs ; 0 ] == 01//02 ^ 0 qed
end math ]"

----------(27.10.06)


"[  math in theory system Q lemma lemma lessMultiplicationLeft(R) says
for all terms meta fx comma meta fy comma meta fz indeed
00 << meta fz infer meta fx << meta fy infer meta fz ** meta fx << meta fz ** meta fy
end lemma end math ]"

"[ math system Q proof of lemma lessMultiplicationLeft(R) reads
any term meta fx comma meta fy comma meta fz end line
line ell a premise 00 << meta fz end line
line ell b premise meta fx << meta fy end line
line ell c because lemma lessMultiplication(R) modus ponens ell a modus ponens ell b indeed
  meta fx ** meta fz << meta fy ** meta fz end line
line ell d because lemma timesCommutativity(R) indeed meta fx ** meta fz == meta fz ** meta fx end line
line ell e because lemma timesCommutativity(R) indeed meta fy ** meta fz == meta fz ** meta fy end line

line ell f because lemma subLessLeft(R) modus ponens ell d modus ponens ell c indeed
  meta fz ** meta fx << meta fy ** meta fz end line
because lemma subLessRight(R) modus ponens ell e modus ponens ell f indeed  
    meta fz ** meta fx << meta fz ** meta fy qed
end math ]"




"[  math in theory system Q lemma lemma negativeToLeft(Less)(R) says
for all terms meta fx comma meta fy comma meta fz indeed
meta fx << meta fy ++ -- meta fz infer meta fx ++ meta fz << meta fy
end lemma end math ]"

"[ math system Q proof of lemma negativeToLeft(Less)(R) reads
any term meta fx comma meta fy comma meta fz end line
line ell a premise meta fx << meta fy ++ -- meta fz end line
line ell b because lemma lessAddition(R) modus ponens ell a indeed 
  meta fx ++ meta fz << meta fy ++ -- meta fz ++ meta fz end line

line ell c because lemma three2threeTerms(R) indeed 
  meta fy ++ -- meta fz ++ meta fz == meta fy ++ meta fz ++ -- meta fz end line
line ell d because lemma x=x+y-y(R) indeed meta fy == meta fy ++ meta fz ++ -- meta fz end line
line ell e because lemma ==Symmetry modus ponens ell d indeed 
  meta fy ++ meta fz ++ -- meta fz == meta fy end line
line ell f because lemma ==Transitivity modus ponens ell c modus ponens ell e indeed
  meta fy ++ -- meta fz ++ meta fz == meta fy end line

because lemma subLessRight(R) modus ponens ell f modus ponens ell b indeed meta fx ++ meta fz << meta fy qed
end math ]"


"[  math in theory system Q lemma lemma negativeToLeft(Less)(1 term)(R) says
for all terms meta fx comma meta fy indeed
 00 << meta fx ++ -- meta fy infer meta fy << meta fx
end lemma end math ]"

"[ math system Q proof of lemma negativeToLeft(Less)(1 term)(R) reads
any term meta fx comma meta fy end line
line ell a premise 00 << meta fx ++ -- meta fy end line
line ell b because lemma negativeToLeft(Less)(R) modus ponens ell a indeed 00 ++ meta fy << meta fx end line
line ell c because lemma plus0Left(R) indeed 00 ++ meta fy == meta fy end line
because lemma subLessLeft(R) modus ponens ell c modus ponens ell b indeed meta fy << meta fx qed
end math ]"


"[  math in theory system Q lemma lemma y-(1/2)(x+y)=(1/2)(y-x)(R) says
for all terms meta fx comma meta fy indeed
  meta fy ++ -- ( 01//02 ** ( meta fx ++ meta fy ) ) == 
  01//02 ** ( meta fy ++ -- meta fx )
end lemma end math ]"

"[ math system Q proof of lemma y-(1/2)(x+y)=(1/2)(y-x)(R) reads
any term meta fx comma meta fy end line
line ell a because lemma distribution(R) indeed 
  01//02 ** ( meta fx ++ meta fy ) == 01//02 ** meta fx ++ 01//02 ** meta fy end line
line ell b because lemma eqNegated(R) modus ponens ell a indeed
  -- ( 01//02 ** ( meta fx ++ meta fy ) ) == 
  -- ( 01//02 ** meta fx ++ 01//02 ** meta fy ) end line
line ell c because lemma eqAdditionLeft(R) modus ponens ell b indeed
  meta fy ++ -- ( 01//02 ** ( meta fx ++ meta fy ) ) == 
  meta fy ++ -- ( 01//02 ** meta fx ++ 01//02 ** meta fy ) end line

line ell d because lemma plusCommutativity(R) indeed 
  01//02 ** meta fx ++ 01//02 ** meta fy == 01//02 ** meta fy ++ 01//02 ** meta fx end line
line ell e because lemma eqNegated(R) modus ponens ell d indeed
  -- ( 01//02 ** meta fx ++ 01//02 ** meta fy ) == 
  -- ( 01//02 ** meta fy ++ 01//02 ** meta fx ) end line

line ell f because lemma -x-y=-(x+y)(R) indeed 
  -- ( 01//02 ** meta fy ) ++ -- ( 01//02 ** meta fx ) == 
  -- ( 01//02 ** meta fy ++ 01//02 ** meta fx ) end line
line ell g because lemma ==Symmetry modus ponens ell f indeed
  -- ( 01//02 ** meta fy ++ 01//02 ** meta fx ) == 
  -- ( 01//02 ** meta fy ) ++ -- ( 01//02 ** meta fx ) end line

line ell h because lemma ==Transitivity modus ponens ell e modus ponens ell g indeed
  -- ( 01//02 ** meta fx ++ 01//02 ** meta fy ) == 
  -- ( 01//02 ** meta fy ) ++ -- ( 01//02 ** meta fx ) end line
line ell i because lemma eqAdditionLeft(R) modus ponens ell h indeed
  meta fy ++ -- ( 01//02 ** meta fx ++ 01//02 ** meta fy ) == 
  meta fy ++ ( -- ( 01//02 ** meta fy ) ++ -- ( 01//02 ** meta fx ) ) end line
line ell big i because lemma plusAssociativity(R) indeed
  meta fy ++   -- ( 01//02 ** meta fy ) ++ -- ( 01//02 ** meta fx ) ==
  meta fy ++ ( -- ( 01//02 ** meta fy ) ++ -- ( 01//02 ** meta fx ) ) end line
line ell big j because lemma ==Symmetry modus ponens ell big i indeed     
  meta fy ++ ( -- ( 01//02 ** meta fy ) ++ -- ( 01//02 ** meta fx ) ) ==
  meta fy ++   -- ( 01//02 ** meta fy ) ++ -- ( 01//02 ** meta fx ) end line

line ell j because lemma (1/2)x+(1/2)x=x(R) indeed 
  01//02 ** meta fy ++ 01//02 ** meta fy == meta fy end line
line ell k because lemma positiveToRight(Eq)(R) modus ponens ell j indeed 
  01//02 ** meta fy == meta fy ++ -- ( 01//02 ** meta fy ) end line
line ell l because lemma ==Symmetry modus ponens ell k indeed 
  meta fy ++ -- ( 01//02 ** meta fy ) == 01//02 ** meta fy end line
line ell m because 1rule adhoc eqAddition(R) modus ponens ell l indeed
  meta fy ++ -- ( 01//02 ** meta fy ) ++ -- ( 01//02 ** meta fx ) == 
  01//02 ** meta fy ++ -- ( 01//02 ** meta fx ) end line

line ell n because lemma distributionOut(Minus)(R) indeed
  01//02 ** meta fy ++ -- ( 01//02 ** meta fx ) == 01//02 ** ( meta fy ++ -- meta fx ) end line

because lemma ==Transitivity6
  modus ponens ell c modus ponens ell i modus ponens ell big j modus ponens ell m modus ponens ell n indeed 
  meta fy ++ -- ( 01//02 ** ( meta fx ++ meta fy ) ) ==
  01//02 ** ( meta fy ++ -- meta fx ) qed
end math ]"


"[ math in theory system Q lemma lemma intervalSize(R) indu says
for all terms meta m indeed 
[ us ; meta m ] ++ -- [ xs ; meta m ] == 01//02 ^ meta m imply
[ us ; meta m + 1 ] ++ -- [ xs ; meta m + 1 ] == 01//02 ^ ( meta m + 1 ) 
end lemma end math ]"

"[ math system Q proof of lemma intervalSize(R) indu reads

block any term meta m end line
line ell b premise upperBound( 01//02 ** ( [ xs ; meta m ] ++ [ us ; meta m ] ) , setOfReals ) end line
line ell a premise [ us ; meta m ] ++ -- [ xs ; meta m ] == 01//02 ^ meta m end line
line ell c because 1rule nextUS(upperBound) modus ponens ell b indeed
  [ us ; meta m + 1 ] == 01//02 ** ( [ xs ; meta m ] ++ [ us ; meta m ] ) end line
line ell d because 1rule nextXS(upperBound) modus ponens ell b indeed 
  [ xs ; meta m + 1 ] == [ xs ; meta m ] end line
line ell e because lemma eqNegated(R) modus ponens ell d indeed 
  -- [ xs ; meta m + 1 ] == -- [ xs ; meta m ] end line
line ell f because lemma addEquations(R) modus ponens ell c modus ponens ell e indeed
  [ us ; meta m + 1 ] ++ -- [ xs ; meta m + 1 ] == 
  01//02 ** ( [ xs ; meta m ] ++ [ us ; meta m ] ) ++ -- [ xs ; meta m ] end line
line ell g because lemma (1/2)(x+y)-x=(1/2)(y-x)(R) indeed
  01//02 ** ( [ xs ; meta m ] ++ [ us ; meta m ] ) ++ -- [ xs ; meta m ] ==
  01//02 ** ( [ us ; meta m ] ++ -- [ xs ; meta m ] ) end line
line ell h because lemma ==Transitivity modus ponens ell f modus ponens ell g indeed
  [ us ; meta m + 1 ] ++ -- [ xs ; meta m + 1 ] ==
  01//02 ** ( [ us ; meta m ] ++ -- [ xs ; meta m ] ) end line

line ell i because lemma eqMultiplicationLeft(R) modus ponens ell a indeed
  01//02 ** ( [ us ; meta m ] ++ -- [ xs ; meta m ] ) == 01//02 ** 01//02 ^ meta m end line

line ell l because lemma ==Transitivity modus ponens ell h modus ponens ell i indeed
  [ us ; meta m + 1 ] ++ -- [ xs ; meta m + 1 ] == 01//02 ** 01//02 ^ meta m end line

line ell j because lemma exp(+1)(R) indeed 
  01//02 ^ ( meta m + 1 ) == 01//02 ** 01//02 ^ meta m end line  
line ell k because lemma ==Symmetry modus ponens ell j indeed
  01//02 ** 01//02 ^ meta m == 01//02 ^ ( meta m + 1 ) end line

because lemma ==Transitivity4 modus ponens ell h modus ponens ell i modus ponens ell k indeed
  [ us ; meta m + 1 ] ++ -- [ xs ; meta m + 1 ] == 01//02 ^ ( meta m + 1 ) end line
line ell big a end block

block any term meta m end line
line ell b premise not0 upperBound( 01//02 ** ( [ xs ; meta m ] ++ [ us ; meta m ] ) , setOfReals ) end line
line ell a premise [ us ; meta m ] ++ -- [ xs ; meta m ] == 01//02 ^ meta m end line
line ell c because 1rule nextUS(noUpperBound) modus ponens ell b indeed
  [ us ; meta m + 1 ] == [ us ; meta m ] end line
line ell d because 1rule nextXS(noUpperBound) modus ponens ell b indeed 
  [ xs ; meta m + 1 ] == 01//02 ** ( [ xs ; meta m ] ++ [ us ; meta m ] ) end line
line ell e because lemma eqNegated(R) modus ponens ell d indeed
  -- [ xs ; meta m + 1 ] == -- ( 01//02 ** ( [ xs ; meta m ] ++ [ us ; meta m ] ) ) end line
line ell f because lemma addEquations(R) modus ponens ell c modus ponens ell e indeed
  [ us ; meta m + 1 ] ++ -- [ xs ; meta m + 1 ] == 
  [ us ; meta m ] ++ -- ( 01//02 ** ( [ xs ; meta m ] ++ [ us ; meta m ] ) ) end line
line ell g because lemma y-(1/2)(x+y)=(1/2)(y-x)(R) indeed
  [ us ; meta m ] ++ -- ( 01//02 ** ( [ xs ; meta m ] ++ [ us ; meta m ] ) ) == 
  01//02 ** ( [ us ; meta m ] ++ -- [ xs ; meta m ] ) end line
line ell h because lemma ==Transitivity modus ponens ell f modus ponens ell g indeed
  [ us ; meta m + 1 ] ++ -- [ xs ; meta m + 1 ] ==
  01//02 ** ( [ us ; meta m ] ++ -- [ xs ; meta m ] ) end line

line ell i because lemma eqMultiplicationLeft(R) modus ponens ell a indeed
  01//02 ** ( [ us ; meta m ] ++ -- [ xs ; meta m ] ) == 01//02 ** 01//02 ^ meta m end line

line ell j because lemma exp(+1)(R) indeed 
  01//02 ^ ( meta m + 1 ) == 01//02 ** 01//02 ^ meta m end line  
line ell k because lemma ==Symmetry modus ponens ell j indeed
  01//02 ** 01//02 ^ meta m == 01//02 ^ ( meta m + 1 ) end line

because lemma ==Transitivity4 modus ponens ell h modus ponens ell i modus ponens ell k indeed
  [ us ; meta m + 1 ] ++ -- [ xs ; meta m + 1 ] == 01//02 ^ ( meta m + 1 ) end line
line ell big b end block

any term meta m end line

line ell a because 1rule deduction modus ponens ell big a indeed
  upperBound( 01//02 ** ( [ xs ; meta m ] ++ [ us ; meta m ] ) , setOfReals ) imply
  [ us ; meta m ] ++ -- [ xs ; meta m ] == 01//02 ^ meta m imply
  [ us ; meta m + 1 ] ++ -- [ xs ; meta m + 1 ] == 01//02 ^ ( meta m + 1 ) end line
line ell b because 1rule deduction modus ponens ell big b indeed
  not0 upperBound( 01//02 ** ( [ xs ; meta m ] ++ [ us ; meta m ] ) , setOfReals ) imply
  [ us ; meta m ] ++ -- [ xs ; meta m ] == 01//02 ^ meta m imply
  [ us ; meta m + 1 ] ++ -- [ xs ; meta m + 1 ] == 01//02 ^ ( meta m + 1 ) end line

because prop lemma from negations modus ponens ell a modus ponens ell b indeed 
  [ us ; meta m ] ++ -- [ xs ; meta m ] == 01//02 ^ meta m imply
  [ us ; meta m + 1 ] ++ -- [ xs ; meta m + 1 ] == 01//02 ^ ( meta m + 1 ) qed
end math ]"




"[ math in theory system Q lemma lemma intervalSize(R) says
for all terms meta m indeed
[ us ; meta m ] ++ -- [ xs ; meta m ] == 01//02 ^ meta m 
end lemma end math ]"


"[ math system Q proof of lemma intervalSize(R) reads
any term meta m end line
line ell a because lemma intervalSize(R) base indeed 
  [ us ; 0 ] ++ -- [ xs ; 0 ] == 01//02 ^ 0  end line
line ell b because lemma intervalSize(R) indu indeed
  [ us ; meta m ] ++ -- [ xs ; meta m ] == 01//02 ^ meta m imply
  [ us ; meta m + 1 ] ++ -- [ xs ; meta m + 1 ] == 01//02 ^ ( meta m + 1 ) end line

because lemma induction modus ponens ell a modus ponens ell b indeed 
[ us ; meta m ] ++ -- [ xs ; meta m ] == 01//02 ^ meta m qed
end math ]"



"[  math in theory system Q lemma lemma XSlessUS(R) says
for all terms meta m indeed
[ xs ; meta m ] << [ us ; meta m ]
end lemma end math ]"

"[ math system Q proof of lemma XSlessUS(R) reads 
any term meta m end line
line ell a because lemma intervalSize(R) indeed 
  [ us ; meta m ] ++ -- [ xs ; meta m ] == 01//02 ^ meta m end line
line ell big a because lemma ==Symmetry modus ponens ell a indeed
  01//02 ^ meta m == [ us ; meta m ] ++ -- [ xs ; meta m ] end line
line ell b because lemma 0<1/2(R) indeed 00 << 01//02 end line
line ell c because lemma positiveBase(R) modus ponens ell b indeed 00 << 01//02 ^ meta m end line
line ell d because lemma subLessRight(R) modus ponens ell big a modus ponens ell c indeed
  00 << [ us ; meta m ] ++ -- [ xs ; meta m ] end line
because lemma negativeToLeft(Less)(1 term)(R) indeed [ xs ; meta m ] << [ us ; meta m ] qed
end math ]"


"[  math in theory system Q lemma lemma USdecreasing(+1)(R) says
for all terms meta m indeed
[ us ; meta m + 1 ] <<== [ us ; meta m ]
end lemma end math ]"


"[ math system Q proof of lemma USdecreasing(+1)(R) reads
block any term meta m end line
line ell a premise 
  upperBound( 01//02 ** ( [ xs ; meta m ] ++ [ us ; meta m ] ) , setOfReals ) end line
line ell b because 1rule nextUS(upperBound) modus ponens ell a indeed
  [ us ; meta m + 1 ] == 01//02 ** ( [ xs ; meta m ] ++ [ us ; meta m ] ) end line

line ell big b because lemma ==Symmetry modus ponens ell b indeed
  01//02 ** ( [ xs ; meta m ] ++ [ us ; meta m ] ) == [ us ; meta m + 1 ] end line

line ell c because lemma XSlessUS(R) indeed [ xs ; meta m ] << [ us ; meta m ] end line
line ell d because lemma lessAddition(R) modus ponens ell c indeed
  [ xs ; meta m ] ++ [ us ; meta m ] << [ us ; meta m ] ++ [ us ; meta m ] end line

line ell e because lemma 0<1/2(R) indeed 00 << 01//02 end line
line ell f because lemma lessMultiplicationLeft(R) modus ponens ell e modus ponens ell d indeed
  01//02 ** ( [ xs ; meta m ] ++ [ us ; meta m ] ) << 
  01//02 ** ( [ us ; meta m ] ++ [ us ; meta m ] ) end line

line ell g because lemma subLessLeft(R) modus ponens ell big b modus ponens ell f indeed
  [ us ; meta m + 1 ] << 01//02 ** ( [ us ; meta m ] ++ [ us ; meta m ] ) end line

line ell h because lemma distribution(R) indeed
  01//02 ** ( [ us ; meta m ] ++ [ us ; meta m ] ) == 
  01//02 ** [ us ; meta m ] ++ 01//02 ** [ us ; meta m ] end line
line ell i because lemma (1/2)x+(1/2)x=x(R) indeed
  01//02 ** [ us ; meta m ] ++ 01//02 ** [ us ; meta m ] == [ us ; meta m ] end line
line ell j because lemma ==Transitivity modus ponens ell h modus ponens ell i indeed
  01//02 ** ( [ us ; meta m ] ++ [ us ; meta m ] ) == [ us ; meta m ] end line

line ell k because lemma subLessRight(R) modus ponens ell j modus ponens ell g indeed
  [ us ; meta m + 1 ] << [ us ; meta m ] end line
because lemma lessLeq(R) modus ponens ell k indeed [ us ; meta m + 1 ] <<== [ us ; meta m ] end line
line ell big a end block

block any term meta m end line
line ell a premise not0 upperBound( 01//02 ** ( [ xs ; meta m ] ++ [ us ; meta m ] ) , setOfReals ) end line
line ell b because 1rule nextUS(noUpperBound) modus ponens ell a indeed 
  [ us ; meta m + 1 ] == [ us ; meta m ] end line
because lemma eqLeq(R) modus ponens ell b indeed [ us ; meta m + 1 ] <<== [ us ; meta m ] end line
line ell big b end block

any term meta m end line
line ell a because 1rule deduction modus ponens ell big a indeed
  upperBound( 01//02 ** ( [ xs ; meta m ] ++ [ us ; meta m ] ) , setOfReals ) imply
  [ us ; meta m + 1 ] <<== [ us ; meta m ] end line
line ell b because 1rule deduction modus ponens ell big b indeed
  not0 upperBound( 01//02 ** ( [ xs ; meta m ] ++ [ us ; meta m ] ) , setOfReals ) imply
  [ us ; meta m + 1 ] <<== [ us ; meta m ] end line


because prop lemma from negations modus ponens ell a modus ponens ell b indeed 
  [ us ; meta m + 1 ] <<== [ us ; meta m ] qed
end math ]"

----------(28.10.06)

"[  math in theory system Q lemma lemma 1<=x+1(N) says
for all terms meta m indeed
Nat( meta m ) endorse 1 <= meta m + 1
end lemma end math ]"

"[ math system Q proof of lemma 1<=x+1(N) reads
any term meta m end line
line ell a side condition Nat( meta m ) end line
line ell b because axiom nonnegative(N) modus probans ell a indeed 0 <= meta m end line
line ell c because lemma leqAddition modus ponens ell b indeed 0 + 1 <= meta m + 1 end line
line ell d because lemma plus0Left indeed 0 + 1 = 1 end line
because lemma subLeqLeft modus ponens ell d modus ponens ell c indeed 1 <= meta m + 1 qed
end math ]"


"[  math in theory system Q lemma lemma expUnbounded base says
0 + 1 <= 2 ^ 0
end lemma end math ]"

"[ math system Q proof of lemma expUnbounded base reads
line ell a because lemma expZero exact indeed 2 ^ 0 = 1 end line
line ell b because lemma eqSymmetry modus ponens ell a indeed 1 = 2 ^ 0 end line
line ell c because lemma plus0Left indeed 0 + 1 = 1 end line
line ell d because lemma eqTransitivity modus ponens ell c modus ponens ell b indeed 0 + 1 = 2 ^ 0 end line
because lemma eqLeq modus ponens ell d indeed 0 + 1 <= 2 ^ 0 qed
end math ]"


"[  math in theory system Q lemma lemma expUnbounded indu says
for all terms meta m indeed
meta m + 1 <= 2 ^ meta m imply
meta m + 1 + 1 <= 2 ^ ( meta m + 1 )
end lemma end math ]"

"[ math system Q proof of lemma expUnbounded indu reads
block any term meta m end line
line ell a premise meta m + 1 <= 2 ^ meta m end line
line ell big b because lemma 0<2 indeed 0 < 2 end line
line ell big c because lemma lessLeq modus ponens ell big b indeed 0 <= 2 end line
line ell b because lemma leqMultiplicationLeft modus ponens ell big c modus ponens ell a indeed
  2 * ( meta m + 1 ) <= 2 * 2 ^ meta m end line

line ell c because lemma exp(+1) indeed 2 ^ ( meta m + 1 ) = 2 * 2 ^ meta m end line
line ell d because lemma eqSymmetry modus ponens ell c indeed 2 * 2 ^ meta m = 2 ^ ( meta m + 1 ) end line
line ell e because lemma subLeqRight modus ponens ell d modus ponens ell b indeed 
  2 * ( meta m + 1 ) <= 2 ^ ( meta m + 1 ) end line

line ell f because lemma x+x=2*x indeed ( meta m + 1 ) + ( meta m + 1 ) = 2 * ( meta m + 1 ) end line
line ell g because axiom leqReflexivity indeed meta m + 1 <= meta m + 1 end line
line ell h because lemma 1<=x+1(N) indeed 1 <= meta m + 1 end line
line ell i because lemma addEquations(Leq) modus ponens ell g modus ponens ell h indeed
  meta m + 1 + 1 <= ( meta m + 1 ) + ( meta m + 1 ) end line
line ell j because  lemma subLeqRight modus ponens ell f modus ponens ell i indeed
  meta m + 1 + 1 <= 2 * ( meta m + 1 ) end line
because lemma leqTransitivity modus ponens ell j modus ponens ell e indeed
  meta m + 1 + 1 <= 2 ^ ( meta m + 1 ) end line
line ell big a end block

any term meta m end line

because 1rule deduction modus ponens ell big a indeed 
  meta m + 1 <= 2 ^ meta m imply meta m + 1 + 1 <= 2 ^ ( meta m + 1 ) qed
end math ]"


"[  math in theory system Q lemma lemma expUnbounded says
for all terms meta m comma meta n indeed
exist0 meta n indeed meta m < 2 ^ meta n
end lemma end math ]"

"[ math system Q proof of lemma expUnbounded reads
any term meta m comma meta n end line
line ell a because lemma expUnbounded base indeed 0 + 1 <= 2 ^ 0 end line
line ell b because lemma expUnbounded indu indeed 
  meta m + 1 <= 2 ^ meta m imply
  meta m + 1 + 1 <= 2 ^ ( meta m + 1 ) end line
line ell c because lemma induction modus ponens ell a modus ponens ell b indeed
  meta m + 1 <= 2 ^ meta m end line

line ell d because lemma leqPlus1 modus ponens ell c indeed meta m + 1 < 2 ^ meta m + 1 end line
line ell e because lemma positiveToRight(Less) modus ponens ell d indeed 
  meta m < 2 ^ meta m + 1 - 1 end line
line ell f because lemma x=x+y-y indeed 2 ^ meta m = 2 ^ meta m + 1 - 1 end line
line ell g because lemma eqSymmetry modus ponens ell f indeed 2 ^ meta m + 1 - 1 = 2 ^ meta m end line
line ell h because lemma subLessRight modus ponens ell g modus ponens ell e indeed 
  meta m < 2 ^ meta m end line

because pred lemma intro exist at meta m modus ponens ell g indeed 
  exist0 meta n indeed meta m < 2 ^ meta n qed
end math ]"

---------(30.10.06)


"[  math in theory system Q lemma lemma nonzeroProduct(2)(R) says
for all terms meta fx comma meta fy indeed
meta fx ** meta fy !!== 00 infer meta fy !!== 00
end lemma end math ]"

"[ math system Q proof of lemma nonzeroProduct(2)(R) reads
block any term meta fx comma meta fy end line
line ell a premise meta fy == 00 end line
line ell b because lemma eqMultiplicationLeft(R) modus ponens ell a indeed 
  meta fx ** meta fy == meta fx ** 00 end line
line ell c because lemma x*0=0(R) indeed meta fx ** 00 == 00 end line
because lemma ==Transitivity modus ponens ell b modus ponens ell c indeed meta fx ** meta fy == 00 end line
line ell big a end block

any term meta fx comma meta fy end line

line ell a because 1rule deduction modus ponens ell big a indeed 
  meta fy == 00 imply meta fx ** meta fy == 00 end line

line ell b premise meta fx ** meta fy !!== 00 end line

because prop lemma mt modus ponens ell a modus ponens ell b indeed meta fy !!== 00 qed
end math ]"

"[  math in theory system Q lemma lemma positiveNonzero(R) says
for all terms meta fx indeed
00 << meta fx infer meta fx !!== 00 
end lemma end math ]"

"[ math system Q proof of lemma positiveNonzero(R) reads
any term meta fx end line
line ell a premise 00 << meta fx end line
line ell c because lemma lessNeq(R) modus ponens ell a indeed 00 !!== meta fx end line

because lemma !!==Symmetry modus ponens ell c indeed meta fx !!== 00 qed
end math ]"


"[  math in theory system Q lemma lemma nonreciprocalToRight(Eq)(1 term)(R) says
for all terms meta fx comma meta fy indeed
meta fx ** meta fy == 01 infer
meta fx == 01// meta fy 
end lemma end math ]"


"[ math system Q proof of lemma nonreciprocalToRight(Eq)(1 term)(R) reads
any term meta fx comma meta fy end line
line ell a premise meta fx ** meta fy == 01 end line
line ell b because lemma eqMultiplication(R) modus ponens ell a indeed
  meta fx ** meta fy ** 01// meta fy == 01 ** 01// meta fy end line

line ell c because lemma 0<1(R) indeed 00 << 01 end line
line ell d because lemma positiveNonzero(R) modus ponens ell c indeed 01 !!== 00 end line
line ell e because lemma ==Symmetry modus ponens ell a indeed 01 == meta fx ** meta fy end line
line ell f because lemma subNeqLeft(R) modus ponens ell e modus ponens ell d indeed 
  meta fx ** meta fy !!== 00 end line
line ell g because lemma nonzeroProduct(2)(R) modus ponens ell f indeed meta fy !!== 00 end line

line ell h because lemma x=x*y*(1/y)(R) modus ponens ell g indeed 
  meta fx == meta fx ** meta fy ** 01// meta fy end line

line ell i because lemma times1Left(R) indeed 01 ** 01// meta fy == 01// meta fy end line

because lemma ==Transitivity4 modus ponens ell h modus ponens ell b modus ponens ell i indeed 
  meta fx == 01// meta fy qed
end math ]"


"[  math in theory system Q lemma lemma expNonzero base says
for all terms meta fx indeed
meta fx ^ 0 !!== 00
end lemma end math ]"

"[ math system Q proof of lemma expNonzero base reads
any term meta fx end line
line ell b because lemma expZero exact(R) indeed meta fx ^ 0 == 01 end line
line ell c because lemma ==Symmetry modus ponens ell b indeed 01 == meta fx ^ 0 end line
line ell d because lemma 0<1(R) indeed 00 << 01 end line
line ell e because lemma positiveNonzero(R) modus ponens ell d indeed 01 !!== 00 end line

because lemma subNeqLeft(R) modus ponens ell c modus ponens ell e indeed meta fx ^ 0 !!== 00 qed
end math ]"



"[  math in theory system Q lemma lemma expNonzero indu says
for all terms meta n comma meta fx indeed
meta fx !!== 00 infer 
meta fx ^ meta n !!== 00 imply meta fx ^ ( meta n + 1 ) !!== 00
end lemma end math ]"

"[ math system Q proof of lemma expNonzero indu reads
block any term meta n comma meta fx end line
line ell a premise meta fx !!== 00 end line
line ell b premise meta fx ^ meta n !!== 00 end line

line ell c because lemma exp(+1)(R) indeed meta fx ^ ( meta n + 1 ) == meta fx ** meta fx ^ meta n end line
line ell d because lemma ==Symmetry modus ponens ell c indeed 
  meta fx ** meta fx ^ meta n == meta fx ^ ( meta n + 1 ) end line

line ell e because lemma nonzeroFactors(R) modus ponens ell a modus ponens ell b indeed
  meta fx ** meta fx ^ meta n !!== 00 end line
because lemma subNeqLeft(R) modus ponens ell d modus ponens ell e indeed 
  meta fx ^ ( meta n + 1 ) !!== 00 end line
line ell big a end block

any term meta n comma meta fx end line
line ell a because 1rule deduction modus ponens ell big a indeed
  meta fx !!== 00 imply meta fx ^ meta n !!== 00 imply meta fx ^ ( meta n + 1 ) !!== 00 end line

line ell b premise meta fx !!== 00 end line

because 1rule mp modus ponens ell a modus ponens ell b indeed 
  meta fx ^ meta n !!== 00 imply meta fx ^ ( meta n + 1 ) !!== 00 qed
end math ]"


"[  math in theory system Q lemma lemma expNonzero says
for all terms meta n comma meta fx indeed
meta fx !!== 00 infer meta fx ^ meta n !!== 00
end lemma end math ]"

"[ math system Q proof of lemma expNonzero reads
any term meta n comma meta fx end line
line ell a premise meta fx !!== 00 end line
line ell b because lemma expNonzero base indeed meta fx ^ 0 !!== 00 end line
line ell c because lemma expNonzero indu modus ponens ell a indeed
  meta fx ^ meta n !!== 00 imply meta fx ^ ( meta n + 1 ) !!== 00 end line

because lemma induction modus ponens ell b modus ponens ell c indeed meta fx ^ meta n !!== 00 qed
end math ]"

----------(31.10.06)

"[ math in theory system Q lemma lemma multiplyEquations(R) says
for all terms meta fx comma meta fy comma meta fz comma meta fu indeed
meta fx == meta fy infer meta fz == meta fu infer meta fx ** meta fz == meta fy ** meta fu
end lemma end math ]"

"[ math system Q proof of lemma multiplyEquations(R) reads
any term meta fx comma meta fy comma meta fz comma meta fu end line
line ell a premise meta fx == meta fy end line
line ell b premise meta fz == meta fu end line
line ell c because lemma eqMultiplication(R) modus ponens ell a indeed 
  meta fx ** meta fz == meta fy ** meta fz end line
line ell d because lemma eqMultiplicationLeft(R) modus ponens ell b indeed
  meta fy ** meta fz == meta fy ** meta fu end line
because lemma ==Transitivity modus ponens ell c modus ponens ell d indeed 
  meta fx ** meta fz == meta fy ** meta fu qed end math ]"



"[  math in theory system Q lemma lemma expNonzero(2) says
for all terms meta n indeed 02 ^ meta n !!== 00 
end lemma end math ]"

"[ math system Q proof of lemma expNonzero(2) reads
any term meta n end line
line ell a because lemma 0<2(R) indeed 00 << 02 end line
line ell b because lemma positiveNonzero(R) modus ponens ell a indeed 02 !!== 00 end line
because lemma expNonzero modus ponens ell b indeed 02 ^ meta n !!== 00 qed
end math ]"


"[  math in theory system Q lemma lemma halfBase base says
01//02 ^ 0 == 01// 02 ^ 0 
end lemma end math ]"

"[ math system Q proof of lemma halfBase base reads
line ell a because lemma expZero exact(R) indeed 01//02 ^ 0 == 01 end line

line ell b because lemma expZero exact(R) indeed 02 ^ 0 == 01 end line
line ell big a because lemma times1Left(R) indeed 01 ** 02 ^ 0 == 02 ^ 0 end line
line ell big b because lemma ==Transitivity modus ponens ell big a modus ponens ell b indeed 
  01 ** 02 ^ 0 == 01 end line
line ell c because lemma nonreciprocalToRight(Eq)(1 term)(R) modus ponens ell big b
  indeed 01 == 01// 02 ^ 0 end line

because lemma ==Transitivity modus ponens ell a modus ponens ell c indeed
  01//02 ^ 0 == 01// 02 ^ 0 qed
end math ]"


"[ math in theory system Q lemma lemma halfBase indu says
for all terms meta m indeed 
01//02 ^ meta m == 01// 02 ^ meta m imply
01//02 ^ ( meta m + 1 ) == 01// 02 ^ ( meta m + 1 )
end lemma end math ]"

"[ math system Q proof of lemma halfBase indu reads
block any term meta m end line
line ell a premise 01//02 ^ meta m == 01// 02 ^ meta m end line

line ell b because lemma exp(+1)(R) indeed 01//02 ^ ( meta m + 1 ) == 01//02 ** 01//02 ^ meta m end line
line ell c because lemma exp(+1)(R) indeed 02 ^ ( meta m + 1 ) == 02 ** 02 ^ meta m end line
line ell d because lemma multiplyEquations(R) modus ponens ell b modus ponens ell c indeed
  01//02 ^ ( meta m + 1 ) ** 02 ^ ( meta m + 1 ) == 
  01//02 ** 01//02 ^ meta m ** ( 02 ** 02 ^ meta m ) end line


line ell s because lemma eqMultiplicationLeft(R) modus ponens ell a indeed
  01//02 ** 01//02 ^ meta m == 01//02 ** 01// 02 ^ meta m end line
line ell u because lemma eqMultiplication(R) modus ponens ell s indeed
 01//02 ** 01//02 ^ meta m ** ( 02 ** 02 ^ meta m ) == 
 01//02 ** 01// 02 ^ meta m ** ( 02 ** 02 ^ meta m ) end line


line ell n because lemma timesAssociativity(R) indeed
  01//02 ** 01// 02 ^ meta m ** ( 02 ** 02 ^ meta m ) == 
  01//02 ** ( 01// 02 ^ meta m ** ( 02 ** 02 ^ meta m ) ) end line


line ell f because lemma 0<2(R) indeed 00 << 02 end line
line ell g because lemma positiveNonzero(R) modus ponens ell f indeed 02 !!== 00 end line
line ell h because lemma expNonzero modus ponens ell g indeed 02 ^ meta m !!== 00 end line
line ell i because lemma reciprocal(R) modus ponens ell h indeed  
  02 ^ meta m ** 01// 02 ^ meta m == 01 end line
line ell j because lemma eqMultiplicationLeft(R) modus ponens ell i indeed
  02 ** ( 02 ^ meta m ** 01// 02 ^ meta m ) == 02 ** 01 end line
line ell k because lemma timesCommutativity(R) indeed  
  01// 02 ^ meta m ** ( 02 ** 02 ^ meta m ) == 02 ** 02 ^ meta m ** 01// 02 ^ meta m end line
line ell big k because lemma timesAssociativity(R) indeed
  02 ** 02 ^ meta m ** 01// 02 ^ meta m == 02 ** ( 02 ^ meta m ** 01// 02 ^ meta m ) end line
line ell l because lemma times1(R) indeed 02 ** 01 == 02 end line

line ell m because lemma ==Transitivity5 modus ponens ell k modus ponens ell big k modus ponens ell j 
  modus ponens ell l indeed 01// 02 ^ meta m ** ( 02 ** 02 ^ meta m ) == 02 end line
line ell o because lemma eqMultiplicationLeft(R) modus ponens ell m indeed
  01//02 ** ( 01// 02 ^ meta m ** ( 02 ** 02 ^ meta m ) ) == 01//02 ** 02 end line


line ell p because lemma timesCommutativity(R) indeed 01//02 ** 02 == 02 ** 01//02 end line
line ell q because lemma reciprocal(R) modus ponens ell g indeed 02 ** 01//02 == 01 end line

line ell r because lemma ==Transitivity6 modus ponens ell d modus ponens ell u modus ponens ell n 
  modus ponens ell o modus ponens ell p indeed 
  01//02 ^ ( meta m + 1 ) ** 02 ^ ( meta m + 1 ) == 02 ** 01//02 end line
line ell v because lemma ==Transitivity modus ponens ell r modus ponens ell q indeed
  01//02 ^ ( meta m + 1 ) ** 02 ^ ( meta m + 1 ) == 01 end line
because lemma nonreciprocalToRight(Eq)(1 term)(R) modus ponens ell v indeed
  01//02 ^ ( meta m + 1 ) == 01// 02 ^ ( meta m + 1 ) end line
line ell big a end block

any term meta m end line

because 1rule deduction modus ponens ell big a indeed 
  01//02 ^ meta m == 01// 02 ^ meta m imply 01//02 ^ ( meta m + 1 ) == 01// 02 ^ ( meta m + 1 ) qed
end math ]"

"[  math in theory system Q lemma lemma halfBase says
for all terms meta m indeed 
01//02 ^ meta m == 01// 02 ^ meta m
end lemma end math ]"

"[ math system Q proof of lemma halfBase reads
any term meta m end line
line ell a because lemma halfBase base indeed 01//02 ^ 0 == 01// 02 ^ 0 end line
line ell b because lemma halfBase indu indeed 
  01//02 ^ meta m == 01// 02 ^ meta m imply 01//02 ^ ( meta m + 1 ) == 01// 02 ^ ( meta m + 1 ) end line
because lemma induction modus ponens ell a modus ponens ell b indeed 01//02 ^ meta m == 01// 02 ^ meta m qed
end math ]"

-------(2.11.06)

"[  math in theory system Q lemma lemma three2threeFactors(R) says
for all terms meta fx comma meta fy comma meta fz indeed
meta fx ** meta fy ** meta fz == meta fx ** meta fz ** meta fy
end lemma end math ]"

"[ math system Q proof of lemma three2threeFactors(R) reads
any term meta fx comma meta fy comma meta fz end line
line ell a because lemma timesCommutativity(R) indeed meta fy ** meta fz == meta fz ** meta fy end line
line ell b because lemma three2twoFactors(R) modus ponens ell a indeed
  meta fx ** meta fy ** meta fz == meta fx ** ( meta fz ** meta fy ) end line

line ell c because lemma timesAssociativity(R) indeed
  meta fx ** meta fz ** meta fy == meta fx ** parenthesis meta fz ** meta fy end parenthesis end line 
line ell d because lemma ==Symmetry modus ponens ell c indeed
  meta fx ** ( meta fz ** meta fy ) == meta fx ** meta fz ** meta fy end line

because lemma ==Transitivity modus ponens ell b modus ponens ell d indeed  
  meta fx ** meta fy ** meta fz == meta fx ** meta fz ** meta fy qed
end math ]"

"[  math in theory system Q lemma lemma x*y=zBackwards(R) says
for all terms meta fx comma meta fy comma meta fz indeed
meta fx ** meta fy == meta fz infer meta fz == meta fy ** meta fx
end lemma end math ]"

"[ math system Q proof of lemma x*y=zBackwards(R) reads
any term meta fx comma meta fy comma meta fz end line
line ell a premise meta fx ** meta fy == meta fz end line
line ell b because lemma timesCommutativity(R) indeed meta fy ** meta fx == meta fx ** meta fy end line
line ell c because lemma ==Transitivity modus ponens ell b modus ponens ell a indeed 
  meta fy ** meta fx == meta fz end line

because lemma ==Symmetry modus ponens ell c indeed meta fz == meta fy ** meta fx qed
end math ]"


XX 0<1/2 er et specialtilfaelde
"[  math in theory system Q lemma lemma positiveInverted(R) says
for all terms meta fx indeed 00 << meta fx infer 00 << 01// meta fx
end lemma end math ]"

"[ math system Q proof of lemma positiveInverted(R) reads
any term meta fx end line
line ell a premise 00 << meta fx end line
line ell b because lemma lessLeq(R) modus ponens ell a indeed 00 <<== meta fx end line
line ell d because lemma positiveNonzero(R) modus ponens ell a indeed meta fx !!== 00 end line
line ell e because lemma 0<1(R) indeed 00 << 01 end line
line ell f because lemma x*0=0(R) indeed meta fx ** 00 == 00 end line

line ell g because lemma x*y=zBackwards(R) indeed 00 == 00 ** meta fx end line

line ell j because lemma subLessLeft(R) modus ponens ell g modus ponens ell e indeed 
  00 ** meta fx << 01 end line
line ell k because lemma reciprocal(R) modus ponens ell d indeed meta fx ** 01// meta fx == 01 end line

line ell l because lemma x*y=zBackwards(R) modus ponens ell k indeed 01 == 01// meta fx ** meta fx end line
line ell m because lemma subLessRight(R) modus ponens ell l modus ponens ell j indeed
  00 ** meta fx << 01// meta fx ** meta fx end line
because lemma lessDivision(R) modus ponens ell b modus ponens ell m indeed 00 << 01// meta fx qed
end math ]"


"[  math in theory system Q lemma lemma reciprocalToRight(Less)(R) says
for all terms meta fx comma meta fy comma meta fz indeed
00 << meta fy infer meta fx ** 01// meta fy << meta fz infer meta fx << meta fz ** meta fy
end lemma end math ]"

"[ math system Q proof of lemma reciprocalToRight(Less)(R) reads
any term meta fx comma meta fy comma meta fz end line
line ell a premise 00 << meta fy end line
line ell b premise meta fx ** 01// meta fy << meta fz end line
line ell c because lemma lessMultiplication(R) modus ponens ell a modus ponens ell b indeed 
  meta fx ** 01// meta fy ** meta fy << meta fz ** meta fy end line

line ell d because lemma positiveNonzero(R) modus ponens ell a indeed meta fy !!== 00 end line
line ell e because lemma x=x*y*(1/y)(R) modus ponens ell d indeed 
  meta fx == meta fx ** meta fy ** 01// meta fy end line
line ell f because lemma three2threeFactors(R) indeed 
  meta fx ** meta fy ** 01// meta fy == meta fx ** 01// meta fy ** meta fy end line
line ell g because lemma ==Transitivity modus ponens ell e modus ponens ell f indeed
  meta fx == meta fx ** 01// meta fy ** meta fy end line
line ell h because lemma ==Symmetry modus ponens ell g indeed
  meta fx ** 01// meta fy ** meta fy == meta fx end line

because lemma subLessLeft(R) modus ponens ell h modus ponens ell c indeed meta fx << meta fz ** meta fy qed
end math ]"



"[  math in theory system Q lemma lemma reciprocalToRight(Less)(1 term)(R) says
for all terms meta fy comma meta fz indeed
00 << meta fy infer 01// meta fy << meta fz infer 01 << meta fz ** meta fy
end lemma end math ]"

"[ math system Q proof of lemma reciprocalToRight(Less)(1 term)(R) reads

any term meta fy comma meta fz end line
line ell a premise 00 << meta fy end line
line ell b premise 01// meta fy << meta fz end line
line ell c because lemma times1Left(R) indeed 01 ** 01// meta fy == 01// meta fy end line
line ell d because lemma ==Symmetry modus ponens ell c indeed 01// meta fy == 01 ** 01// meta fy end line
line ell e because lemma subLessLeft(R) modus ponens ell d modus ponens ell b indeed 
  01 ** 01// meta fy << meta fz end line

because lemma reciprocalToRight(Less)(R) modus ponens ell a modus ponens ell e indeed 
  01 << meta fz ** meta fy qed
end math ]"


"[  math in theory system Q lemma lemma nonreciprocalToLeft(Less)(R) says
for all terms meta fx comma meta fy comma meta fz indeed
00 << meta fz infer meta fx << meta fy ** meta fz infer meta fx ** 01// meta fz << meta fy
end lemma end math ]"

"[ math system Q proof of lemma nonreciprocalToLeft(Less)(R) reads
any term meta fx comma meta fy comma meta fz end line
line ell a premise 00 << meta fz end line
line ell b premise meta fx << meta fy ** meta fz end line  

line ell c because lemma positiveInverted(R) modus ponens ell a indeed 00 << 01// meta fz end line
line ell d because lemma lessMultiplication(R) modus ponens ell c modus ponens ell b indeed
  meta fx ** 01// meta fz << meta fy ** meta fz ** 01// meta fz end line

line ell e because lemma positiveNonzero(R) modus ponens ell a indeed meta fz !!== 00 end line
line ell f because lemma x=x*y*(1/y)(R) modus ponens ell e indeed 
  meta fy == meta fy ** meta fz ** 01// meta fz end line
line ell g because lemma ==Symmetry modus ponens ell f indeed
  meta fy ** meta fz ** 01// meta fz == meta fy end line

because lemma subLessRight(R) modus ponens ell g modus ponens ell d indeed 
  meta fx ** 01// meta fz << meta fy qed
end math ]"



"[ math in theory system Q lemma lemma 1<x*y(R) says
for all terms meta fx comma meta fy indeed
00 << meta fy infer 01 << meta fx ** meta fy infer 01// meta fy << meta fx
end lemma end math ]"

"[ math system Q proof of lemma 1<x*y(R) reads
any term meta fx comma meta fy end line
line ell a premise 00 << meta fy end line
line ell b premise 01 << meta fx ** meta fy end line
line ell c because lemma nonreciprocalToLeft(Less)(R) modus ponens ell a modus ponens ell b indeed
  01 ** 01// meta fy << meta fx end line
line ell d because lemma times1Left(R) indeed 01 ** 01// meta fy == 01// meta fy end line 

because lemma subLessLeft(R) modus ponens ell d modus ponens ell c indeed 01// meta fy << meta fx qed
end math ]"



"[ math in theory system Q lemma lemma switchFactors(1/x<y)(R) says
for all terms meta fx comma meta fy indeed
00 << meta fx infer 00 << meta fy infer
01// meta fx << meta fy infer 01// meta fy << meta fx
end lemma end math ]" 

"[ math system Q proof of lemma switchFactors(1/x<y)(R) reads
any term meta fx comma meta fy end line
line ell a premise 00 << meta fx end line
line ell b premise 00 << meta fy end line
line ell c premise 01// meta fx << meta fy end line
line ell d because lemma reciprocalToRight(Less)(1 term)(R) modus ponens ell a modus ponens ell c indeed
  01 << meta fy ** meta fx end line
line ell e because lemma timesCommutativity(R) indeed meta fy ** meta fx == meta fx ** meta fy end line
line ell f because lemma subLessRight(R) modus ponens ell e modus ponens ell d indeed 
  01 << meta fx ** meta fy end line
because lemma 1<x*y(R) modus ponens ell b modus ponens ell f indeed
  01// meta fy << meta fx qed
end math ]"


"[  math in theory system Q lemma lemma smallHalving says
for all terms meta n comma meta fx indeed
00 << meta fx infer exist0 meta n indeed 01//02 ^ meta n << meta fx
end lemma end math ]"

"[ math system Q proof of lemma smallHalving reads
block any term meta n comma meta fx end line
line ell a premise 00 << meta fx end line
line ell b premise 01// meta fx << 02 ^ meta n end line

line ell c because lemma 0<2(R) indeed 00 << 02 end line
line ell d because lemma positiveBase(R) modus ponens ell c indeed 00 << 02 ^ meta n end line



line ell e because lemma switchFactors(1/x<y)(R) modus ponens ell a modus ponens ell d 
  modus ponens ell b indeed 01// 02 ^ meta n << meta fx end line

line ell f because lemma halfBase indeed 01//02 ^ meta n == 01// 02 ^ meta n end line
line ell g because lemma ==Symmetry modus ponens ell f indeed 01// 02 ^ meta n == 01//02 ^ meta n end line
line ell h because lemma subLessLeft(R) modus ponens ell g modus ponens ell e 
  indeed 01//02 ^ meta n << meta fx end line
because pred lemma intro exist at meta n modus ponens ell h indeed
  exist0 meta n indeed 01//02 ^ meta n << meta fx end line
line ell big a end block

any term meta n comma meta fx end line

line ell a because 1rule deduction modus ponens ell big a indeed
  00 << meta fx imply 
  01// meta fx << 02 ^ meta n imply exist0 meta n indeed 01//02 ^ meta n << meta fx end line
line ell b premise 00 << meta fx end line
line ell c because 1rule expUnbounded indeed exist0 meta n indeed  01// meta fx << 02 ^ meta n end line


line ell d because 1rule mp modus ponens ell a modus ponens ell b indeed 
  01// meta fx << 02 ^ meta n imply exist0 meta n indeed 01//02 ^ meta n << meta fx end line
because pred lemma exist mp modus ponens ell d modus ponens ell c indeed
  exist0 meta n indeed 01//02 ^ meta n << meta fx qed
end math ]"

"[  math in theory system Q lemma lemma intervalSize(anyPositive) says
for all terms meta m comma meta fep indeed
00 << meta fep infer exist0 meta m indeed [ us ; meta m ] ++ -- [ xs ; meta m ] << meta fep
end lemma end math ]"

"[ math system Q proof of lemma intervalSize(anyPositive) reads
block any term meta m comma meta fep end line
"";line ell a premise 00 << meta fep end line
line ell b premise 01//02 ^ meta m << meta fep end line
line ell c because lemma intervalSize(R) indeed 
  [ us ; meta m ] ++ -- [ xs ; meta m ] == 01//02 ^ meta m end line
line ell d because lemma ==Symmetry modus ponens ell c indeed
  01//02 ^ meta m == [ us ; meta m ] ++ -- [ xs ; meta m ] end line
line ell e because lemma subLessLeft(R) modus ponens ell d modus ponens ell b indeed
  [ us ; meta m ] ++ -- [ xs ; meta m ] << meta fep end line
because pred lemma intro exist at meta m modus ponens ell e indeed
  exist0 meta m indeed [ us ; meta m ] ++ -- [ xs ; meta m ] << meta fep end line
line ell big a end block

any term meta m comma meta fep end line

line ell a because 1rule deduction modus ponens ell big a indeed 01//02 ^ meta m << meta fep imply 
  exist0 meta m indeed [ us ; meta m ] ++ -- [ xs ; meta m ] << meta fep end line

line ell b premise 00 << meta fep end line
line ell c because lemma smallHalving modus ponens ell b indeed 
  exist0 meta m indeed 01//02 ^ meta m << meta fep end line

because pred lemma exist mp modus ponens ell a modus ponens ell c indeed 
  exist0 meta m indeed [ us ; meta m ] ++ -- [ xs ; meta m ] << meta fep qed
end math ]"


"[  math in theory system Q lemma lemma USdecreasing(+n) base says
for all terms meta m indeed
[ us ; meta m + 0 ] <<== [ us ; meta m ] 
end lemma end math ]"

"[ math system Q proof of lemma USdecreasing(+n) base reads
any term meta m end line
line ell a because axiom plus0 indeed meta m + 0 = meta m end line
line ell b because lemma sameSeries modus ponens ell a indeed 
  [ us ; meta m + 0 ] == [ us ; meta m ] end line
because lemma eqLeq(R) modus ponens ell b indeed [ us ; meta m + 0 ] <<== [ us ; meta m ] qed
end math ]"

"[  math in theory system Q lemma lemma USdecreasing(+n) indu says
for all terms meta m comma meta n indeed
[ us ; meta m + meta n ] <<== [ us ; meta m ] imply
[ us ; meta m + ( meta n + 1 ) ] <<== [ us ; meta m ]
end lemma end math ]"

"[ math system Q proof of lemma USdecreasing(+n) indu reads
block any term meta m comma meta n end line
line ell a premise [ us ; meta m + meta n ] <<== [ us ; meta m ] end line
line ell b because lemma USdecreasing(+1)(R) indeed 
  [ us ; meta m + meta n + 1 ] <<== [ us ; meta m + meta n ] end line

line ell c because axiom plusAssociativity indeed meta m + meta n + 1 = meta m + ( meta n + 1 ) end line
line ell d because lemma sameSeries modus ponens ell c indeed
  [ us ; meta m + meta n + 1 ] == [ us ; meta m + ( meta n + 1 ) ] end line
line ell e because lemma subLeqLeft(R) modus ponens ell d modus ponens ell b indeed
  [ us ; meta m + ( meta n + 1 ) ] <<== [ us ; meta m + meta n ] end line

because lemma leqTransitivity(R) modus ponens ell e modus ponens ell a indeed
  [ us ; meta m + ( meta n + 1 ) ] <<== [ us ; meta m ] end line
line ell big a end block

any term meta m comma meta n end line

because 1rule deduction modus ponens ell big a indeed [ us ; meta m + meta n ] <<== [ us ; meta m ] imply 
  [ us ; meta m + ( meta n + 1 ) ] <<== [ us ; meta m ] qed
end math ]"

"[  math in theory system Q lemma lemma USdecreasing(+n) says
for all terms meta m comma meta n indeed
[ us ; meta m + meta n ] <<== [ us ; meta m ] 
end lemma end math ]"

"[ math system Q proof of lemma USdecreasing(+n) reads
any term meta m comma meta n end line
line ell a because lemma USdecreasing(+n) base indeed [ us ; meta m + 0 ] <<== [ us ; meta m ] end line
line ell b because lemma USdecreasing(+n) indu indeed [ us ; meta m + meta n ] <<== [ us ; meta m ] imply 
  [ us ; meta m + ( meta n + 1 ) ] <<== [ us ; meta m ] end line
because lemma induction modus ponens ell a modus ponens ell b indeed 
  [ us ; meta m + meta n ] <<== [ us ; meta m ] qed
end math ]"


"[  math in theory system Q lemma lemma USdecreasing says
for all terms meta m comma meta m1 comma meta n indeed ""; XX ekstra variable
meta m <= meta n infer [ us ; meta n ] <<== [ us ; meta m ]
end lemma end math ]"

"[ math system Q proof of lemma USdecreasing reads
block any term meta m comma meta m1 comma meta n end line
line ell b premise meta m + meta m1 = meta n end line
line ell c because lemma sameSeries modus ponens ell b indeed 
  [ us ; meta m + meta m1 ] == [ us ; meta n ] end line
line ell d because lemma USdecreasing(+n) indeed [ us ; meta m + meta m1 ] <<== [ us ; meta m ] end line
because lemma subLeqLeft(R) modus ponens ell c modus ponens ell d indeed 
  [ us ; meta n ] <<== [ us ; meta m ] end line
line ell big a end block

any term meta m comma meta m1 comma meta n end line
line ell a because 1rule deduction modus ponens ell big a indeed
  meta m + meta m1 = meta n imply [ us ; meta n ] <<== [ us ; meta m ] end line
line ell b premise meta m <= meta n end line
line ell c because 1rule fromLeq(Advanced)(N) modus ponens ell b indeed exist0 meta m1 indeed 
  meta m + meta m1 = meta n end line
because pred lemma exist mp modus ponens ell a modus ponens ell c indeed 
  [ us ; meta n ] <<== [ us ; meta m ] qed
end math ]"

-------(2.11.06)


"[  math in theory system Q lemma lemma leqAdditionLeft(R) says
for all terms meta fx comma meta fy comma meta fz indeed
meta fx <<== meta fy infer meta fz ++ meta fx <<== meta fz ++ meta fy
end lemma end math ]"

"[ math system Q proof of lemma leqAdditionLeft(R) reads
any term meta fx comma meta fy comma meta fz end line
line ell a premise meta fx <<== meta fy end line
line ell b because lemma leqAddition(R) modus ponens ell a indeed 
  meta fx ++ meta fz <<== meta fy ++ meta fz end line

line ell c because lemma plusCommutativity(R) indeed meta fx ++ meta fz == meta fz ++ meta fx end line
line ell d because lemma plusCommutativity(R) indeed meta fy ++ meta fz == meta fz ++ meta fy end line

line ell e because lemma subLeqLeft(R) modus ponens ell c modus ponens ell b indeed 
  meta fz ++ meta fx <<== meta fy ++ meta fz end line

because lemma subLeqRight(R) modus ponens ell d modus ponens ell e indeed 
  meta fz ++ meta fx <<== meta fz ++ meta fy qed
end math ]"

"[  math in theory system Q lemma lemma toNotLess(R) says
for all terms meta fx comma meta fy indeed
meta fx <<== meta fy infer not0 meta fy << meta fx
end lemma end math ]"

"[ math system Q proof of lemma toNotLess(R) reads
block any term meta fx comma meta fy end line
line ell a premise meta fy << meta fx end line
because lemma fromLess(R) modus ponens ell a indeed not0 meta fx <<== meta fy end line
line ell big a end block

any term meta fx comma meta fy end line

line ell a because 1rule deduction modus ponens ell big a indeed
  meta fy << meta fx imply not0 meta fx <<== meta fy end line
line ell b premise meta fx <<== meta fy end line
line ell c because prop lemma add double neg modus ponens ell b indeed 
  not0 not0 meta fx <<== meta fy end line
because prop lemma mt modus ponens ell a modus ponens ell c indeed not0 meta fy << meta fx qed
end math ]"



"[  math in theory system Q lemma lemma limitOfUSIsLeq says
for all terms meta m comma meta m2 comma meta n comma meta fep comma meta fx indeed ""; XX ekstra variable
limit( meta fx , usF ) infer meta fx <<== [ usF ; meta m ]
end lemma end math ]"

"[ math system Q proof of lemma limitOfUSIsLeq reads
block any term meta m comma meta m2 comma meta n comma meta fx end line
line ell b premise for all meta n indeed ( 00 << meta fx ++ -- [ usF ; meta m ] imply meta m2 <= meta n
  imply |r meta fx ++ -- [ usF ; meta n ] | << meta fx ++ -- [ usF ; meta m ] ) end line
line ell a premise not0 meta fx <<== [ usF ; meta m ] end line
line ell c because lemma a4 at max( meta m , meta m2 ) modus ponens ell b indeed
  00 << meta fx ++ -- [ usF ; meta m ] imply meta m2 <= max( meta m , meta m2 )
  imply |r meta fx ++ -- [ usF ; max( meta m , meta m2 ) ] | << meta fx ++ -- [ usF ; meta m ] end line

line ell d because lemma toLess(R) modus ponens ell a indeed [ usF ; meta m ] << meta fx end line
line ell e because lemma positiveToRight(Less)(1 term)(R) modus ponens ell d indeed
  00 << meta fx ++ -- [ usF ; meta m ] end line
line ell f because lemma leqMax2 indeed meta m2 <= max( meta m , meta m2 ) end line

line ell g because prop lemma mp2 modus ponens ell c modus ponens ell e modus ponens ell f indeed
  |r meta fx ++ -- [ usF ; max( meta m , meta m2 ) ] | << meta fx ++ -- [ usF ; meta m ] end line

line ell h because lemma leqMax1 indeed meta m <= max( meta m , meta m2 ) end line
line ell i because lemma USdecreasing modus ponens ell h indeed
  [ usF ; max( meta m , meta m2 ) ] <<== [ usF ; meta m ] end line
line ell j because lemma leqNegated(R) modus ponens ell i indeed 
  -- [ usF ; meta m ] <<== -- [ usF ; max( meta m , meta m2 ) ] end line
line ell k because lemma leqAdditionLeft(R) modus ponens ell j indeed
  meta fx ++ -- [ usF ; meta m ] <<== meta fx ++ -- [ usF ; max( meta m , meta m2 ) ] end line

line ell l because lemma x<=|x|(R) indeed  
  meta fx ++ -- [ usF ; max( meta m , meta m2 ) ] <<== 
  |r meta fx ++ -- [ usF ; max( meta m , meta m2 ) ] | end line

line ell m because lemma leqTransitivity(R) modus ponens ell k modus ponens ell l indeed
  meta fx ++ -- [ usF ; meta m ] <<== |r meta fx ++ -- [ usF ; max( meta m , meta m2 ) ] | end line
line ell n because lemma toNotLess(R) modus ponens ell m indeed
  not0 |r meta fx ++ -- [ usF ; max( meta m , meta m2 ) ] | << meta fx ++ -- [ usF ; meta m ] end line
because prop lemma from contradiction modus ponens ell g modus ponens ell n indeed 
  not0 not0 meta fx <<== [ usF ; meta m ] end line
line ell big a end block

any term meta m comma meta m2 comma meta n comma meta fep comma meta fx end line

line ell a because 1rule deduction modus ponens ell big a indeed
for all meta n indeed ( 00 << meta fx ++ -- [ usF ; meta m ] imply meta m2 <= meta n
  imply |r meta fx ++ -- [ usF ; meta n ] | << meta fx ++ -- [ usF ; meta m ] ) imply
not0 meta fx <<== [ usF ; meta m ] imply
not0 not0 meta fx <<== [ usF ; meta m ] end line

line ell b premise limit( meta fx , usF ) end line
line ell c because 1rule fromLimit modus ponens ell b indeed
  for all meta fep indeed exist0 meta m2 indeed for all meta n indeed 
  ( 00 << meta fep imply meta m2 <= meta n imply 
  |r meta fx ++ -- [ usF ; meta n ] | << meta fep ) end line
line ell d because lemma a4 at meta fx ++ -- [ usF ; meta m ] modus ponens ell c indeed
   exist0 meta m2 indeed for all meta n indeed 
  ( 00 << meta fx ++ -- [ usF ; meta m ] imply meta m2 <= meta n imply 
  |r meta fx ++ -- [ usF ; meta n ] | << meta fx ++ -- [ usF ; meta m ] ) end line
  
line ell e because pred lemma exist mp modus ponens ell a modus ponens ell d indeed
  not0 meta fx <<== [ usF ; meta m ] imply not0 not0 meta fx <<== [ usF ; meta m ] end line
line ell f because prop lemma imply negation modus ponens ell e indeed 
  not0 not0 meta fx <<== [ usF ; meta m ] end line

because prop lemma remove double neg modus ponens ell f indeed meta fx <<== [ usF ; meta m ] qed
end math ]"


"[  math in theory system Q lemma lemma subtractEquations(Less)(R) says
for all terms meta fx comma meta fy comma meta fz comma meta fu indeed
meta fx ++ meta fy << meta fz ++ meta fu infer meta fy == meta fu infer meta fx << meta fz
end lemma end math ]"

"[ math system Q proof of lemma subtractEquations(Less)(R) reads
any term meta fx comma meta fy comma meta fz comma meta fu end line
line ell a premise meta fx ++ meta fy << meta fz ++ meta fu end line
line ell b premise meta fy == meta fu end line
line ell c because lemma lessAddition(R) modus ponens ell a indeed
  meta fx ++ meta fy ++ -- meta fy << meta fz ++ meta fu ++ -- meta fy end line

line ell d because lemma x=x+y-y(R) indeed meta fx == meta fx ++ meta fy ++ -- meta fy end line
line ell e because lemma ==Symmetry modus ponens ell d indeed 
  meta fx ++ meta fy ++ -- meta fy == meta fx end line
line ell f because lemma subLessLeft(R) modus ponens ell e modus ponens ell c indeed
  meta fx << meta fz ++ meta fu ++ -- meta fy end line

line ell g because lemma positiveToRight(Eq)(1 term)(R) modus ponens ell b indeed 
  00 == meta fu ++ -- meta fy end line
line ell h because lemma ==Symmetry modus ponens ell g indeed meta fu ++ -- meta fy == 00 end line
line ell i because lemma three2twoTerms(R) modus ponens ell h indeed 
  meta fz ++ meta fu ++ -- meta fy == meta fz ++ 00 end line
line ell j because lemma plus0(R) indeed meta fz ++ 00 == meta fz end line
line ell k because lemma ==Transitivity modus ponens ell i modus ponens ell j indeed
  meta fz ++ meta fu ++ -- meta fy == meta fz end line


because lemma subLessRight(R) modus ponens ell k modus ponens ell f indeed meta fx << meta fz qed
end math ]"
 

"[  math in theory system Q lemma lemma subtractEquationsLeft(Less)(R) says
for all terms meta fx comma meta fy comma meta fz comma meta fu indeed
meta fx ++ meta fy << meta fz ++ meta fu infer meta fx == meta fz infer meta fy << meta fu
end lemma end math ]"

"[ math system Q proof of lemma subtractEquationsLeft(Less)(R) reads
any term meta fx comma meta fy comma meta fz comma meta fu end line
line ell a premise meta fx ++ meta fy << meta fz ++ meta fu end line

line ell b premise meta fx == meta fz end line

line ell c because lemma plusCommutativity(R) indeed meta fx ++ meta fy == meta fy ++ meta fx end line
line ell d because lemma subLessLeft(R) modus ponens ell c modus ponens ell a indeed
  meta fy ++ meta fx << meta fz ++ meta fu end line

line ell e because lemma plusCommutativity(R) indeed meta fz ++ meta fu == meta fu ++ meta fz end line
line ell f because lemma subLessRight(R) modus ponens ell e modus ponens ell d indeed
  meta fy ++ meta fx << meta fu ++ meta fz end line

because lemma subtractEquations(Less)(R) modus ponens ell f modus ponens ell b indeed meta fy << meta fu qed
end math ]"

"[  math in theory system Q lemma lemma lessNegated(Negative)(R) says
for all terms meta fx comma meta fy indeed
-- meta fx << -- meta fy infer meta fy << meta fx
end lemma end math ]"

"[ math system Q proof of lemma lessNegated(Negative)(R) reads
any term meta fx comma meta fy end line
line ell a premise -- meta fx << -- meta fy end line
line ell b because lemma lessNegated(R) modus ponens ell a indeed -- -- meta fy << -- -- meta fx end line

line ell c because lemma doubleMinus(R) indeed -- -- meta fy == meta fy end line
line ell d because lemma subLessLeft(R) modus ponens ell c modus ponens ell b indeed 
  meta fy << -- -- meta fx end line

line ell e because lemma doubleMinus(R) indeed -- -- meta fx == meta fx end line
because lemma subLessRight(R) modus ponens ell e modus ponens ell d indeed meta fy << meta fx qed
end math ]"

"[  math in theory system Q lemma prop lemma from negated and (imply) says
for all terms meta a comma meta b indeed
not0 ( meta a and0 meta b ) imply meta a imply not0 meta b
end lemma end math ]"

"[ math system Q proof of prop lemma from negated and (imply) reads
block any term meta a comma meta b end line
line ell a premise not0 ( meta a and0 meta b ) end line
line ell b premise meta a end line
because prop lemma from negated and modus ponens ell a modus ponens ell b indeed not0 meta b end line
line ell big a end block

any term meta a comma meta b end line
because 1rule deduction modus ponens ell big a indeed 
  not0 ( meta a and0 meta b ) imply meta a imply not0 meta b qed 
end math ]"


"[  math in theory system Q lemma prop lemma remove double neg (consequent) says
for all terms meta a comma meta b indeed
meta a imply not0 not0 meta b infer meta a imply meta b
end lemma end math ]"

"[ math system Q proof of prop lemma remove double neg (consequent) reads
block any term meta a comma meta b end line
line ell a premise meta a imply not0 not0 meta b end line
line ell b premise meta a end line
line ell c because 1rule mp modus ponens ell a modus ponens ell b indeed not0 not0 meta b end line
because prop lemma remove double neg modus ponens ell c indeed meta b end line
line ell big a end block

any term meta a comma meta b end line
line ell a because 1rule deduction modus ponens ell big a indeed 
  ( meta a imply not0 not0 meta b ) imply meta a imply meta b end line
line ell b premise meta a imply not0 not0 meta b end line

because 1rule mp modus ponens ell a modus ponens ell b indeed meta a imply meta b qed
end math ]"



"[  math in theory system Q lemma lemma fromNotUpperBound says
for all terms meta fx comma meta fy comma meta fys indeed
not0 upperBound( meta fx , meta fys ) infer 
exist0 meta fy indeed meta fy in0 meta fys and0 meta fx << meta fy
end lemma end math ]"

"[ math system Q proof of lemma fromNotUpperBound reads
block any term meta fx comma meta fy comma meta fys end line
line ell a premise meta fy in0 meta fys imply meta fy <<== meta fx end line
because 1rule toUpperBound modus ponens ell a indeed upperBound( meta fx , meta fys ) end line
line ell big a end block

any term meta fx comma meta fy comma meta fys end line

line ell a because 1rule deduction modus ponens ell big a indeed 
  ( meta fy in0 meta fys imply meta fy <<== meta fx ) imply upperBound( meta fx , meta fys ) end line

line ell b premise not0 upperBound( meta fx , meta fys ) end line

line ell c because prop lemma mt modus ponens ell a modus ponens ell b indeed
  not0 ( meta fy in0 meta fys imply meta fy <<== meta fx ) end line
line ell d because prop lemma from negated imply modus ponens ell c indeed
  meta fy in0 meta fys and0 not0 meta fy <<== meta fx end line

line ell e because prop lemma first conjunct modus ponens ell d indeed meta fy in0 meta fys end line
line ell f because prop lemma second conjunct modus ponens ell d indeed not0 meta fy <<== meta fx end line
line ell g because lemma toLess(R) modus ponens ell f indeed meta fx << meta fy end line
line ell h because prop lemma join conjuncts modus ponens ell e modus ponens ell g indeed
  meta fy in0 meta fys and0 meta fx << meta fy end line

because pred lemma intro exist at meta fy modus ponens ell h indeed 
  exist0 meta fy indeed meta fy in0 meta fys and0 meta fx << meta fy qed
end math ]"

"[  math in theory system Q lemma lemma leqNUB says
for all terms meta fx comma meta fy comma meta fz comma meta fxs indeed ""; XX ekstra variable
not0 upperBound( meta fy , meta fxs ) infer meta fz <<== meta fy infer not0 upperBound( meta fz , meta fxs )
end lemma end math ]"

"[ math system Q proof of lemma leqNUB reads
block any term meta fx comma meta fy comma meta fz comma meta fxs end line
line ell a premise meta fx in0 meta fxs and0 meta fy << meta fx end line ""; fromNotUpperBound
line ell b premise meta fz <<== meta fy end line
line ell c premise upperBound( meta fz , meta fxs ) end line
 
line ell d because prop lemma first conjunct modus ponens ell a indeed meta fx in0 meta fxs end line
line ell e because prop lemma second conjunct modus ponens ell a indeed meta fy << meta fx end line

line ell f because 1rule fromUpperBound modus ponens ell c modus ponens ell d indeed
  meta fx <<== meta fz end line

line ell g because lemma leqLessTransitivity(R) modus ponens ell b modus ponens ell e indeed
  meta fz << meta fx end line
line ell h because lemma fromLess(R) modus ponens ell g indeed not0 meta fx <<== meta fz end line

because prop lemma from contradiction modus ponens ell f modus ponens ell h indeed
  not0 upperBound( meta fz , meta fxs ) end line
line ell big a end block

any term meta fx comma meta fy comma meta fz comma meta fxs end line
line ell a because 1rule deduction modus ponens ell big a indeed
  meta fx in0 meta fxs and0 meta fy << meta fx imply
  meta fz <<== meta fy imply
  upperBound( meta fz , meta fxs ) imply
  not0 upperBound( meta fz , meta fxs ) end line

line ell b premise not0 upperBound( meta fy , meta fxs ) end line
line ell c premise meta fz <<== meta fy end line

line ell d because lemma fromNotUpperBound modus ponens ell b indeed
  exist0 meta fx indeed meta fx in0 meta fxs and0 meta fy << meta fx end line

line ell e because pred lemma exist mp modus ponens ell a modus ponens ell d indeed
  meta fz <<== meta fy imply upperBound( meta fz , meta fxs ) imply 
  not0 upperBound( meta fz , meta fxs ) end line

line ell f because 1rule mp modus ponens ell e modus ponens ell c indeed 
  upperBound( meta fz , meta fxs ) imply not0 upperBound( meta fz , meta fxs ) end line

because prop lemma imply negation modus ponens ell f indeed not0 upperBound( meta fz , meta fxs ) qed
end math ]"



"[  math in theory system Q lemma lemma USlimitIsLeastUpperBound helper says
for all terms meta m comma meta fy comma meta fz indeed
limit( meta fy , usF ) infer
[ us ; meta m ] ++ -- [ xs ; meta m ] << meta fy ++ -- meta fz imply not0 upperBound( meta fz , setOfFxs )
end lemma end math ]"

"[ math system Q proof of lemma USlimitIsLeastUpperBound helper reads 
block any term meta m comma meta fy comma meta fz end line
line ell a premise limit( meta fy , usF ) end line
""; linie b: intervalSize(anyPositive)
line ell b premise [ us ; meta m ] ++ -- [ xs ; meta m ] << meta fy ++ -- meta fz end line

line ell c because lemma limitOfUSIsLeq modus ponens ell a indeed
  meta fy <<== [ usF ; meta m ] end line
line ell d because lemma leqAddition(R) modus ponens ell c indeed
  meta fy ++ -- [ xs ; meta m ] <<== [ usF ; meta m ] ++ -- [ xs ; meta m ] end line

line ell e because lemma leqLessTransitivity(R) modus ponens ell d modus ponens ell b indeed
  meta fy ++ -- [ xs ; meta m ] << meta fy ++ -- meta fz end line
line ell f because lemma ==Reflexivity indeed meta fy == meta fy end line
line ell g because lemma subtractEquationsLeft(Less)(R) modus ponens ell e modus ponens ell f indeed
  -- [ xs ; meta m ] << -- meta fz end line
line ell h because lemma lessNegated(Negative)(R) modus ponens ell g indeed
  meta fz << [ xs ; meta m ] end line
line ell i because lemma lessLeq(R) modus ponens ell h indeed
  meta fz <<== [ xs ; meta m ] end line

line ell j because axiom XSisNotUpperBound indeed not0 upperBound( [ xs ; meta m ] , setOfFxs ) end line

because lemma leqNUB modus ponens ell j modus ponens ell i indeed
  not0 upperBound( meta fz , setOfFxs ) end line
line ell big a end block

any term meta m comma meta fy comma meta fz end line

line ell a because 1rule deduction modus ponens ell big a indeed
  limit( meta fy , usF ) imply
  [ us ; meta m ] ++ -- [ xs ; meta m ] << meta fy ++ -- meta fz imply 
  not0 upperBound( meta fz , setOfFxs ) end line
line ell b premise limit( meta fy , usF ) end line

because 1rule mp modus ponens ell a modus ponens ell b indeed
  [ us ; meta m ] ++ -- [ xs ; meta m ] << meta fy ++ -- meta fz imply 
  not0 upperBound( meta fz , setOfFxs ) qed
end math ]"


"[  math in theory system Q lemma lemma USlimitIsLeastUpperBound says
for all terms meta m comma meta fy comma meta fz indeed ""; XX ekstra variable
limit( meta fy , usF ) infer leastUpperBound( meta fy , setOfFxs )
end lemma end math ]"

"[ math system Q proof of lemma USlimitIsLeastUpperBound reads
block any term meta m comma meta fy comma meta fz end line
line ell a premise limit( meta fy , usF ) end line
line ell b premise upperBound( meta fz , setOfFxs ) and0 not0 meta fy <<== meta fz end line
line ell big b because prop lemma first conjunct modus ponens ell b indeed 
  upperBound( meta fz , setOfFxs ) end line
line ell big c because prop lemma second conjunct modus ponens ell b indeed
  not0 meta fy <<== meta fz end line
line ell c because lemma toLess(R) modus ponens ell big c indeed meta fz << meta fy end line

line ell d because lemma positiveToRight(Less)(1 term)(R) modus ponens ell c indeed 
  00 << meta fy ++ -- meta fz end line
line ell e because lemma intervalSize(anyPositive) modus ponens ell d indeed
  exist0 meta m indeed [ us ; meta m ] ++ -- [ xs ; meta m ] << meta fy ++ -- meta fz end line
  
line ell f because lemma USlimitIsLeastUpperBound helper modus ponens ell a indeed
  [ us ; meta m ] ++ -- [ xs ; meta m ] << meta fy ++ -- meta fz imply 
  not0 upperBound( meta fz , setOfFxs ) end line

line ell g because pred lemma exist mp modus ponens ell f modus ponens ell e indeed 
  not0 upperBound( meta fz , setOfFxs ) end line

because prop lemma from contradiction modus ponens ell big b modus ponens ell g indeed
  not0 ( upperBound( meta fz , setOfFxs ) and0 not0 meta fy <<== meta fz ) end line
line ell big a end block

any term meta m comma meta fy comma meta fz end line

line ell a because 1rule deduction modus ponens ell big a indeed
  limit( meta fy , usF ) imply upperBound( meta fz , setOfFxs ) and0 not0 meta fy <<== meta fz imply
  not0 ( upperBound( meta fz , setOfFxs ) and0 not0 meta fy <<== meta fz ) end line
line ell b premise limit( meta fy , usF ) end line
line ell c because 1rule mp modus ponens ell a modus ponens ell b indeed
  upperBound( meta fz , setOfFxs ) and0 not0 meta fy <<== meta fz imply
  not0 ( upperBound( meta fz , setOfFxs ) and0 not0 meta fy <<== meta fz ) end line
line ell d because prop lemma imply negation modus ponens ell c indeed
  not0 ( upperBound( meta fz , setOfFxs ) and0 not0 meta fy <<== meta fz ) end line
line ell e because prop lemma from negated and (imply) indeed
 not0 ( upperBound( meta fz , setOfFxs ) and0 not0 meta fy <<== meta fz ) imply
 upperBound( meta fz , setOfFxs ) imply not0 not0 meta fy <<== meta fz  end line
line ell big e because 1rule mp modus ponens ell e modus ponens ell d indeed
 upperBound( meta fz , setOfFxs ) imply not0 not0 meta fy <<== meta fz  end line
line ell f because prop lemma remove double neg (consequent) modus ponens ell big e indeed
  upperBound( meta fz , setOfFxs ) imply meta fy <<== meta fz  end line

line ell g because lemma USlimitIsUpperBound modus ponens ell b indeed 
  upperBound( meta fy , setOfFxs ) end line 

because 1rule toLeastUpperBound modus ponens ell g modus ponens ell f indeed 
  leastUpperBound( meta fy , setOfFxs ) qed
end math ]"

---------(4.11.06)

"[  math in theory system Q lemma pred lemma exist mp3 says
for all terms meta v1 comma meta v2 comma meta v3 comma meta a comma meta b comma meta c comma meta d indeed
meta a imply meta b imply meta c imply meta d infer
exist0 meta v1 indeed meta a infer exist0 meta v2 indeed meta b infer exist0 meta v3 indeed meta c 
infer meta d
end lemma end math ]"

"[ math system Q proof of pred lemma exist mp3 reads
any term meta v1 comma meta v2 comma meta v3 comma meta a comma meta b comma meta c comma meta d end line
line ell a premise meta a imply meta b imply meta c imply meta d end line
line ell b premise exist0 meta v1 indeed meta a end line
line ell c premise exist0 meta v2 indeed meta b end line
line ell d premise exist0 meta v3 indeed meta c end line

line ell e because pred lemma exist mp2 modus ponens ell a modus ponens ell b modus ponens ell c indeed
  meta c imply meta d end line

because pred lemma exist mp modus ponens ell e modus ponens ell d indeed meta d qed
end math ]"

"[  math in theory system Q lemma lemma greaterPositive(N) says
for all terms meta m comma meta x indeed
Nat( meta m ) endorse 
meta m < meta x infer 0 < meta x
end lemma end math ]"

"[ math system Q proof of lemma greaterPositive(N) reads
any term meta m comma meta x end line
line ell a side condition Nat( meta m ) end line
line ell b premise meta m < meta x end line
line ell c because axiom nonnegative(N) modus probans ell a indeed 0 <= meta m end line
because lemma leqLessTransitivity modus ponens ell c modus ponens ell b indeed 0 < meta x qed
end math ]"

"[  math in theory system Q lemma lemma ysFClose helper says
for all terms meta m comma meta n comma meta ep indeed
  1/ meta m < meta ep imply meta n < meta m imply | [ ysF ; meta m ] - [ xsF ; meta m ] | < meta ep 
end lemma end math ]"


"[ math system Q proof of lemma ysFClose helper reads
block any term meta m comma meta n comma meta ep end line
line ell big c premise 1/ meta m < meta ep end line
line ell big b premise meta n < meta m end line
line ell big d because lemma greaterPositive(N) modus ponens ell big b indeed 0 < meta m end line
line ell a because axiom ysFLess indeed [ ysF ; meta m ] < [ xsF ; meta m ] + 1/ meta m end line
line ell b because axiom plusCommutativity indeed 
  [ xsF ; meta m ] + 1/ meta m = 1/ meta m + [ xsF ; meta m ] end line
line ell c because lemma subLessRight modus ponens ell b modus ponens ell a indeed
  [ ysF ; meta m ] < 1/ meta m + [ xsF ; meta m ] end line
line ell d because lemma positiveToLeft(Less) modus ponens ell c indeed
  [ ysF ; meta m ] - [ xsF ; meta m ] < 1/ meta m end line

line ell e because lemma positiveInverted modus ponens ell big d indeed 0 < 1/ meta m end line
line ell f because lemma positiveNegated modus ponens ell e indeed - 1/ meta m < 0 end line

line ell g because axiom ysFGreater indeed [ xsF ; meta m ] < [ ysF ; meta m ] end line
line ell h because lemma positiveToRight(Less)(1 term) modus ponens ell g indeed
  0 < [ ysF ; meta m ] - [ xsF ; meta m ] end line
line ell i because lemma lessTransitivity modus ponens ell f modus ponens ell h indeed
  - 1/ meta m < [ ysF ; meta m ] - [ xsF ; meta m ] end line
line ell j because lemma toNumericalLess modus ponens ell i modus ponens ell d indeed 
  | [ ysF ; meta m ] - [ xsF ; meta m ] | < 1/ meta m end line

because lemma lessTransitivity modus ponens ell j modus ponens ell big c indeed
  | [ ysF ; meta m ] - [ xsF ; meta m ] | < meta ep end line
line ell big a end block

any term meta m comma meta n comma meta ep end line

because 1rule deduction modus ponens ell big a indeed  
  1/ meta m < meta ep imply meta n < meta m imply | [ ysF ; meta m ] - [ xsF ; meta m ] | < meta ep qed
end math ]"

"[  math in theory system Q lemma lemma ysFClose says
for all terms meta m comma meta n comma meta ep indeed
  for all meta ep indeed exist0 meta n indeed for all meta m indeed 
  ( 0 < meta ep imply meta n < meta m imply | [ ysF ; meta m ] - [ xsF ; meta m ] | < meta ep )
end lemma end math ]"

"[ math system Q proof of lemma ysFClose reads
block any term meta m comma meta n comma meta ep end line
line ell a premise 0 < meta ep end line
line ell b premise meta n < meta m end line

line ell c because lemma ysFClose helper indeed 
  1/ meta m < meta ep imply meta n < meta m imply | [ ysF ; meta m ] - [ xsF ; meta m ] | < meta ep end line

line ell d because 1rule smallInverse modus ponens ell a indeed 
  exist0 meta m indeed 1/ meta m < meta ep end line

line ell d because pred lemma exist mp modus ponens ell c modus ponens ell d indeed
  meta n < meta m imply | [ ysF ; meta m ] - [ xsF ; meta m ] | < meta ep end line
because 1rule mp modus ponens ell d modus ponens ell b indeed 
  | [ ysF ; meta m ] - [ xsF ; meta m ] | < meta ep end line
line ell big a end block

any term meta m comma meta n comma meta ep end line

line ell a because 1rule deduction modus ponens ell big a indeed
  0 < meta ep imply meta n < meta m imply | [ ysF ; meta m ] - [ xsF ; meta m ] | < meta ep end line

line ell b because 1rule gen modus ponens ell a indeed for all meta m indeed ( 
  0 < meta ep imply meta n < meta m imply | [ ysF ; meta m ] - [ xsF ; meta m ] | < meta ep ) end line

line ell c because pred lemma intro exist at meta n modus ponens ell b indeed 
  exist0 meta n indeed for all meta m indeed ( 
  0 < meta ep imply meta n < meta m imply | [ ysF ; meta m ] - [ xsF ; meta m ] | < meta ep ) end line

because 1rule gen modus ponens ell c indeed 
  for all meta ep indeed exist0 meta n indeed for all meta m indeed ( 
  0 < meta ep imply meta n < meta m imply | [ ysF ; meta m ] - [ xsF ; meta m ] | < meta ep ) qed
end math ]"


line ell a because 1rule deduction modus ponens ell big a indeed 
  1/ meta m < meta ep imply meta n < meta m imply 
  | [ ysF ; meta m ] - [ xsF ; meta m ] | < meta ep end line

line ell b premise 0 < meta ep end line



"[  math in theory system Q lemma lemma ysFCauchy helper says
for all terms meta m1 comma meta m2 comma meta n1 comma meta n2 comma meta n3 comma meta ep indeed
for all meta m1 indeed ( 
  0 < 1/3 * meta ep imply meta n1 < meta m1 imply 
  | [ ysF ; meta m1 ] - [ xsF ; meta m1 ] | < 1/3 * meta ep ) imply
for all meta m1 comma meta m2 indeed (
  0 < 1/3 * meta ep imply meta n2 <= meta m1 imply meta n2 <= meta m2 imply 
  | [ xsF ; meta m1 ] - [ xsF ; meta m2 ] | < 1/3 * meta ep ) imply
for all meta m2 indeed (
  0 < 1/3 * meta ep imply meta n3 < meta m2 imply 
  | [ ysF ; meta m2 ] - [ xsF ; meta m2 ] | < 1/3 * meta ep ) imply
0 < meta ep imply
max( max( meta n1 , meta n2 ) , meta n3 ) < meta m1 imply
max( max( meta n1 , meta n2 ) , meta n3 ) < meta m2 imply
| [ ysF ; meta m1 ] - [ ysF ; meta m2 ] | < meta ep
end lemma end math ]"

"[ math system Q proof of lemma ysFCauchy helper reads
block any term meta m1 comma meta m2 comma meta n1 comma meta n2 comma meta n3 comma meta ep end line
line ell b premise for all meta m1 indeed ( 
  0 < 1/3 * meta ep imply meta n1 < meta m1 imply 
  | [ ysF ; meta m1 ] - [ xsF ; meta m1 ] | < 1/3 * meta ep ) end line
line ell c premise for all meta m1 comma meta m2 indeed ( ""; xs er cauchy
  0 < 1/3 * meta ep imply meta n2 <= meta m1 imply meta n2 <= meta m2 imply 
  | [ xsF ; meta m1 ] - [ xsF ; meta m2 ] | < 1/3 * meta ep ) end line
line ell d premise for all meta m2 indeed (
  0 < 1/3 * meta ep imply meta n3 < meta m2 imply 
  | [ ysF ; meta m2 ] - [ xsF ; meta m2 ] | < 1/3 * meta ep ) end line

line ell a premise 0 < meta ep end line
line ell e premise max( max( meta n1 , meta n2 ) , meta n3 ) < meta m1 end line
line ell f premise max( max( meta n1 , meta n2 ) , meta n3 ) < meta m2 end line

line ell g because lemma leqMax1 indeed meta n1 <= max( meta n1 , meta n2 ) end line
line ell h because lemma leqMax1 indeed 
  max( meta n1 , meta n2 ) <= max( max( meta n1 , meta n2 ) , meta n3 ) end line
line ell i because lemma leqTransitivity modus ponens ell g modus ponens ell h indeed
  meta n1 <= max( max( meta n1 , meta n2 ) , meta n3 ) end line
line ell j because lemma leqLessTransitivity modus ponens ell i modus ponens ell e indeed
  meta n1 < meta m1 end line
line ell k because lemma 0<1/3 indeed 0 < 1/3 end line
line ell l because lemma positiveFactors modus ponens ell k modus ponens ell a indeed 
  0 < 1/3 * meta ep end line
line ell m because lemma a4 at meta m1 modus ponens ell b indeed
  0 < 1/3 * meta ep imply meta n1 < meta m1 imply 
  | [ ysF ; meta m1 ] - [ xsF ; meta m1 ] | < 1/3 * meta ep end line
line ell n because prop lemma mp2 modus ponens ell m modus ponens ell l modus ponens ell j indeed 
  | [ ysF ; meta m1 ] - [ xsF ; meta m1 ] | < 1/3 * meta ep end line

line ell o because lemma leqMax2 indeed meta n2 <= max( meta n1 , meta n2 ) end line
line ell p because lemma leqTransitivity modus ponens ell o modus ponens ell h indeed
  meta n2 <= max( max( meta n1 , meta n2 ) , meta n3 ) end line
line ell q because lemma leqLessTransitivity modus ponens ell p modus ponens ell e indeed
  meta n2 < meta m1 end line
line ell r because lemma lessLeq modus ponens ell q indeed meta n2 <= meta m1 end line
line ell s because lemma leqLessTransitivity modus ponens ell p modus ponens ell f indeed
  meta n2 < meta m2 end line
line ell t because lemma lessLeq modus ponens ell s indeed meta n2 <= meta m2 end line
line ell u because lemma a4 at meta m1 modus ponens ell c indeed
  for all meta m2 indeed ( 
  0 < 1/3 * meta ep imply meta n2 <= meta m1 imply meta n2 <= meta m2 imply 
  | [ xsF ; meta m1 ] - [ xsF ; meta m2 ] | < 1/3 * meta ep ) end line
line ell v because lemma a4 at meta m2 modus ponens ell u indeed
  0 < 1/3 * meta ep imply meta n2 <= meta m1 imply meta n2 <= meta m2 imply 
  | [ xsF ; meta m1 ] - [ xsF ; meta m2 ] | < 1/3 * meta ep  end line
line ell w because prop lemma mp3 modus ponens ell v modus ponens ell l modus ponens ell r
  modus ponens ell t indeed   
  | [ xsF ; meta m1 ] - [ xsF ; meta m2 ] | < 1/3 * meta ep end line


line ell x because lemma leqMax2 indeed meta n3 <= max( max( meta n1 , meta n2 ) , meta n3 ) end line
line ell y because lemma leqLessTransitivity modus ponens ell x modus ponens ell f indeed
  meta n3 < meta m2 end line 
line ell z because lemma a4 at meta m2 modus ponens ell d indeed
  0 < 1/3 * meta ep imply meta n3 < meta m2 imply 
  | [ ysF ; meta m2 ] - [ xsF ; meta m2 ] | < 1/3 * meta ep end line
line ell big b because prop lemma mp2 modus ponens ell z modus ponens ell l modus ponens ell y indeed
  | [ ysF ; meta m2 ] - [ xsF ; meta m2 ] | < 1/3 * meta ep end line
line ell big c because lemma numericalDifference indeed
  | [ ysF ; meta m2 ] - [ xsF ; meta m2 ] | = | [ xsF ; meta m2 ] - [ ysF ; meta m2 ] | end line
line ell big d because lemma subLessLeft modus ponens ell big c modus ponens ell big b indeed
  | [ xsF ; meta m2 ] - [ ysF ; meta m2 ] | < 1/3 * meta ep end line  

line ell big e because lemma addEquations(Less) modus ponens ell n modus ponens ell w indeed 
  | [ ysF ; meta m1 ] - [ xsF ; meta m1 ] | + | [ xsF ; meta m1 ] - [ xsF ; meta m2 ] | < 
  1/3 * meta ep + 1/3 * meta ep end line
line ell big f because lemma addEquations(Less) modus ponens ell big e modus ponens ell big d indeed
  | [ ysF ; meta m1 ] - [ xsF ; meta m1 ] | + | [ xsF ; meta m1 ] - [ xsF ; meta m2 ] | + 
  | [ xsF ; meta m2 ] - [ ysF ; meta m2 ] | < 1/3 * meta ep + 1/3 * meta ep + 1/3 * meta ep end line

line ell big g because lemma insertTwoMiddleTerms(Numerical) indeed
  | [ ysF ; meta m1 ] - [ ysF ; meta m2 ] | <= | [ ysF ; meta m1 ] - [ xsF ; meta m1 ] | +
  | [ xsF ; meta m1 ] - [ xsF ; meta m2 ] | + | [ xsF ; meta m2 ] - [ ysF ; meta m2 ] | end line
line ell big h because lemma leqLessTransitivity modus ponens ell big g modus ponens ell big f indeed
  | [ ysF ; meta m1 ] - [ ysF ; meta m2 ] | < 1/3 * meta ep + 1/3 * meta ep + 1/3 * meta ep end line

line ell big i because lemma (1/3)x+(1/3)x+(1/3)x=x indeed
  1/3 * meta ep + 1/3 * meta ep + 1/3 * meta ep = meta ep end line
because lemma subLessRight modus ponens ell big i modus ponens ell big h indeed
  | [ ysF ; meta m1 ] - [ ysF ; meta m2 ] | < meta ep end line
line ell big a end block


any term meta m1 comma meta m2 comma meta n1 comma meta n2 comma meta n3 comma meta ep end line
because 1rule deduction modus ponens ell big a indeed
for all meta m1 indeed ( 
  0 < 1/3 * meta ep imply meta n1 < meta m1 imply 
  | [ ysF ; meta m1 ] - [ xsF ; meta m1 ] | < 1/3 * meta ep ) imply
for all meta m1 comma meta m2 indeed (
  0 < 1/3 * meta ep imply meta n2 <= meta m1 imply meta n2 <= meta m2 imply 
  | [ xsF ; meta m1 ] - [ xsF ; meta m2 ] | < 1/3 * meta ep ) imply
for all meta m2 indeed (
  0 < 1/3 * meta ep imply meta n3 < meta m2 imply 
  | [ ysF ; meta m2 ] - [ xsF ; meta m2 ] | < 1/3 * meta ep ) imply
0 < meta ep imply
max( max( meta n1 , meta n2 ) , meta n3 ) < meta m1 imply
max( max( meta n1 , meta n2 ) , meta n3 ) < meta m2 imply
| [ ysF ; meta m1 ] - [ ysF ; meta m2 ] | < meta ep qed
end math ]"


"[  math in theory system Q lemma lemma ysFCauchy says
for all terms meta m1 comma meta m2 comma meta n comma meta n1 comma meta n2 comma meta n3 comma meta ep indeed
""; XX ekstra variable
for all meta ep indeed exist0 meta n indeed for all meta m1 comma meta m2 indeed (
  0 < meta ep imply meta n < meta m1 imply meta n < meta m2 imply 
  | [ ysF ; meta m1 ] - [ ysF ; meta m2 ] | < meta ep )
end lemma end math ]"

"[ math system Q proof of lemma ysFCauchy reads
any term meta m1 comma meta m2 comma meta n comma meta n1 comma meta n2 comma meta n3 comma meta ep end line

line ell d because lemma ysFClose indeed
  for all meta ep indeed exist0 meta n1 indeed for all meta m1 indeed ( 
  0 < meta ep imply meta n1 < meta m1 imply | [ ysF ; meta m1 ] - [ xsF ; meta m1 ] | < meta ep ) end line
line ell e because lemma a4 at 1/3 * meta ep modus ponens ell d indeed
  exist0 meta n1 indeed for all meta m1 indeed ( 0 < 1/3 * meta ep imply meta n1 < meta m1 imply 
  | [ ysF ; meta m1 ] - [ xsF ; meta m1 ] | < 1/3 * meta ep ) end line

line ell f because axiom cauchy indeed 
  for all meta ep indeed exist0 meta n2 indeed for all meta m1 comma meta m2 indeed (
  0 < meta ep imply meta n2 <= meta m1 imply meta n2 <= meta m2 imply 
  | [ xsF ; meta m1 ] - [ xsF ; meta m2 ] | < meta ep ) end line
line ell g because lemma a4 at 1/3 * meta ep modus ponens ell f indeed
  exist0 meta n1 indeed for all meta m1 comma meta m2 indeed (
  0 < 1/3 * meta ep imply meta n2 <= meta m1 imply meta n2 <= meta m2 imply 
  | [ xsF ; meta m1 ] - [ xsF ; meta m2 ] | < 1/3 * meta ep ) end line

line ell h because lemma ysFClose indeed
  for all meta ep indeed exist0 meta n3 indeed for all meta m2 indeed ( 
  0 < meta ep imply meta n3 < meta m2 imply | [ ysF ; meta m2 ] - [ xsF ; meta m2 ] | < meta ep ) end line
line ell i because lemma a4 at 1/3 * meta ep modus ponens ell h indeed
  exist0 meta n3 indeed for all meta m2 indeed ( 0 < 1/3 * meta ep imply meta n3 < meta m2 imply 
  | [ ysF ; meta m2 ] - [ xsF ; meta m2 ] | < 1/3 * meta ep ) end line

line ell j because lemma ysFCauchy helper indeed
for all meta m1 indeed ( 
  0 < 1/3 * meta ep imply meta n1 < meta m1 imply 
  | [ ysF ; meta m1 ] - [ xsF ; meta m1 ] | < 1/3 * meta ep ) imply
for all meta m1 comma meta m2 indeed (
  0 < 1/3 * meta ep imply meta n2 <= meta m1 imply meta n2 <= meta m2 imply 
  | [ xsF ; meta m1 ] - [ xsF ; meta m2 ] | < 1/3 * meta ep ) imply
for all meta m2 indeed (
  0 < 1/3 * meta ep imply meta n3 < meta m2 imply 
  | [ ysF ; meta m2 ] - [ xsF ; meta m2 ] | < 1/3 * meta ep ) imply
0 < meta ep imply
max( max( meta n1 , meta n2 ) , meta n3 ) < meta m1 imply
max( max( meta n1 , meta n2 ) , meta n3 ) < meta m2 imply
| [ ysF ; meta m1 ] - [ ysF ; meta m2 ] | < meta ep end line

line ell k because pred lemma exist mp3 modus ponens ell j modus ponens ell e modus ponens ell g 
  modus ponens ell i indeed 
  0 < meta ep imply
  max( max( meta n1 , meta n2 ) , meta n3 ) < meta m1 imply
  max( max( meta n1 , meta n2 ) , meta n3 ) < meta m2 imply
  | [ ysF ; meta m1 ] - [ ysF ; meta m2 ] | < meta ep end line

line ell l because 1rule gen modus ponens ell k indeed for all meta m2 indeed (
  0 < meta ep imply
  max( max( meta n1 , meta n2 ) , meta n3 ) < meta m1 imply
  max( max( meta n1 , meta n2 ) , meta n3 ) < meta m2 imply
  | [ ysF ; meta m1 ] - [ ysF ; meta m2 ] | < meta ep ) end line
line ell m because 1rule gen modus ponens ell l indeed for all meta m1 comma meta m2 indeed (
  0 < meta ep imply
  max( max( meta n1 , meta n2 ) , meta n3 ) < meta m1 imply
  max( max( meta n1 , meta n2 ) , meta n3 ) < meta m2 imply
  | [ ysF ; meta m1 ] - [ ysF ; meta m2 ] | < meta ep ) end line
line ell n because pred lemma intro exist at max( max( meta n1 , meta n2 ) , meta n3 ) modus ponens ell l
  indeed exist0 meta n indeed for all meta m1 comma meta m2 indeed (
  0 < meta ep imply meta n < meta m1 imply meta n < meta m2 imply
  | [ ysF ; meta m1 ] - [ ysF ; meta m2 ] | < meta ep ) end line

because 1rule gen modus ponens ell n indeed
  for all meta ep indeed exist0 meta n indeed for all meta m1 comma meta m2 indeed (
  0 < meta ep imply meta n < meta m1 imply meta n < meta m2 imply
  | [ ysF ; meta m1 ] - [ ysF ; meta m2 ] | < meta ep ) qed
end math ]"

STOR FYLDER SLUTTER}  ]"

venter----


\appendix

"[ flush left math priority table Priority end table end math end left ]"

\section{Pyk definitioner} \label{sec:pyk}

\begin{flushleft}
"[ math protect Pyk end protect end math ]"
\end{flushleft}

\newpage

\begin{list}{}{
\setlength{\leftmargin}{0em}
\setlength{\itemindent}{0em}
\setlength{\itemsep}{1ex}}

\item "[ math tex define sup  
   as "sup"  end define end math ]" 

\item "[ math tex define var x ^ var y  ""; ^
   as "(#1."n (expARGH!) #2."n)"  end define end math ]"



\item "[ math tex define lemma to!!==  ""; REGELLEMMAER
   as "To!!=="  end define end math ]" 

\item "[ math tex define prop lemma to negated double imply ""; UDSAGNSLOGIK 
   as "ToNegatedDoubleImply"  end define end math ]" 

\item "[ math tex define pred lemma addNegatedAll ""; PRAEDIKATLOGIK  
   as "AddNegatedAll"  end define end math ]" 

\item "[ math tex define pred lemma (A)to(~E~)(Imply)  
   as "(A)to(~E~)(Imply)"  end define end math ]" 

\item "[ math tex define pred lemma (E)to(~A~)(Imply)  
   as "(E)to(~A~)(Imply)"  end define end math ]" 

\item "[ math tex define pred lemma (E~)to(~A)(Imply)  
   as "(E~)to(~A)(Imply)"  end define end math ]" 

\item "[ math tex define pred lemma toNegatedAEA  
   as "ToNegatedAEA "  end define end math ]" 

\item "[ math tex define lemma nonreciprocalToRight(Eq)(1 term) ""; EQ
   as "NonreciprocalToRight(Eq)(1 term)"  end define end math ]" 

\item "[ math tex define lemma distributionOut(Minus)  
   as "DistributionOut(Minus)"  end define end math ]" 
 
\item "[ math tex define lemma positiveToRight(Eq)(1 term)  
   as "PositiveToRight(Eq)(1 term)"  end define end math ]" 

\item "[ math tex define lemma sameSeries(NumDiff)  
   as "SameSeries(NumDiff)"  end define end math ]" 

\item "[ math tex define lemma plusAssociativity(4 terms)  
   as "PlusAssociativity(4 terms)"  end define end math ]" 

\item "[ math tex define lemma nonzeroProduct(2) ""; NEQ 
   as "NonzeroProduct(2)"  end define end math ]"

\item "[ math tex define lemma switchTerms(x<=y-z) ""; TEXLEQ  
   as "SwitchTerms(x<=y-z)"  end define end math ]" 

\item "[ math tex define lemma eqLeq(R) 
   as "eqLeq(R)"  end define end math ]" 

\item "[ math tex define lemma thirdGeqSeries  
   as "ThirdGeqSeries"  end define end math ]" 

\item "[ math tex define lemma negativeToLeft(Leq)
           as "negativeToLeft(Leq)"  end define end math ]"

\item "[ math tex define lemma negativeToLeft(Leq)(1 term)
           as "negativeToLeft(Leq)(1 term)"  end define end math ]"

\item "[ math tex define lemma negativeToLeft(Less)(1 term) ""; LESS  

   as "NegativeToLeft(Less)(1 term)"  end define end math ]" 

\item "[ math tex define lemma (1/2)(x+y)-x=(1/2)(y-x) ""; REGNESTYKKER  
   as "(1/2)(x+y)-x=(1/2)(y-x)"  end define end math ]" 

\item "[ math tex define lemma y-(1/2)(x+y)=(1/2)(y-x) 
   as "y-(1/2)(x+y)=(1/2)(y-x)"  end define end math ]" 

\item "[ math tex define lemma +1IsPositive(N)   ""; NATURLIGE TAL
   as "(+1)IsPositive(N)"  end define end math ]" 

\item "[ math tex define lemma expZero exact ""; EXP 
   as "ExpZero(Exact) "  end define end math ]" 

\item "[ math tex define lemma sameExp base  
   as "SameExp(Base)"  end define end math ]" 

\item "[ math tex define lemma sameExp indu  
   as "SameExp(Indu)"  end define end math ]"

\item "[ math tex define lemma sameExp
   as "SameExp"  end define end math ]"

\item "[ math tex define lemma exp(+1)  
   as "Exp(+1)"  end define end math ]" 

\item "[ math tex define lemma base(1/2)Sum zero exact ""; BASE(1/2)SUM 
   as "BSzero(Exact)"  end define end math ]" 

\item "[ math tex define lemma sameBase(1/2)Sum second base  
   as "SameBS(2)(Base)"  end define end math ]" 

\item "[ math tex define lemma sameBase(1/2)Sum second indu
   as "SameBS(2)(Indu)"  end define end math ]" 

\item "[ math tex define lemma sameBase(1/2)Sum second
   as "SameBS(2)"  end define end math ]" 

\item "[ math tex define lemma base(1/2)Sum(+1)  
   as "BS(+1)"  end define end math ]" 

\item "[ math tex define lemma base(1/2)Sum exact bound base  
   as "BSbound(Exact)(Base)"  end define end math ]" 

\item "[ math tex define lemma base(1/2)Sum exact bound indu
   as "BSbound(Exact)(Indu)"  end define end math ]"

\item "[ math tex define lemma base(1/2)Sum exact bound
   as "BSbound(Exact)"  end define end math ]"

\item "[ math tex define lemma base(1/2)Sum bound  
   as "BSbound"  end define end math ]"

\item "[ math tex define lemma UStelescope zero exact ""; TELESCOPE 
   as "UStelescope(Zero)(Exact)"  end define end math ]" 

\item "[ math tex define lemma sameTelescope second base  
   as "SameTelescope(2)(Base)"  end define end math ]" 

\item "[ math tex define lemma sameTelescope second indu  
   as "SameTelescope(2)(Indu)"  end define end math ]"

\item "[ math tex define lemma sameTelescope second  
   as "SameTelescope(2)"  end define end math ]"

\item "[ math tex define lemma UStelescope(+1)  
   as "UStelescope(+1)"  end define end math ]" 

\item "[ math tex define lemma telescopeNumerical base  
   as "TelescopeNumerical(Base)"  end define end math ]" 

\item "[ math tex define lemma telescopeNumerical indu
   as "TelescopeNumerical(Indu)"  end define end math ]"

\item "[ math tex define lemma telescopeNumerical 
   as "TelescopeNumerical"  end define end math ]"

\item "[ math tex define lemma telescopeBound base  
   as "TelescopeBound(Base)"  end define end math ]" 

\item "[ math tex define lemma telescopeBound indu
   as "TelescopeBound(Indu)"  end define end math ]"

\item "[ math tex define lemma telescopeBound 
   as "TelescopeBound"  end define end math ]"

\item "[ math tex define lemma lessNeq(F) helper ""; F/R 
   as "LessNeq(F)(Helper)"  end define end math ]" 

\item "[ math tex define lemma lessNeq(F)  
   as "LessNeq(F)"  end define end math ]" 

\item "[ math tex define lemma lessNeq(R) 
   as "LessNeq(R)"  end define end math ]" 

\item "[ math tex define lemma intervalSize base ""; SUP	 
   as "IntervalSize(Base)"  end define end math ]" 

\item "[ math tex define lemma intervalSize indu 
   as "IntervalSize(Indu)"  end define end math ]" 

\item "[ math tex define lemma intervalSize  
   as "IntervalSize"  end define end math ]" 

\item "[ math tex define lemma XSlessUS  
   as "XS<US"  end define end math ]" 

\item "[ math tex define lemma positiveBase base  
   as "PositiveBase(Base)"  end define end math ]" 

\item "[ math tex define lemma positiveBase indu
   as "PositiveBase(Indu)"  end define end math ]" 

\item "[ math tex define lemma positiveBase
   as "PositiveBase"  end define end math ]" 

\item "[ math tex define lemma closeUS  
   as "CloseUS"  end define end math ]" 

\item "[ math tex define lemma closeUS(n+1)  
   as "CloseUS(n+1)"  end define end math ]" 

\item "[ math tex define lemma induction ""; REGELLEMMAER 
   as "Induction"  end define end math ]" 

\item "[ math tex define lemma leqAntisymmetry  
   as "leqAntisymmetry"  end define end math ]"

\item "[ math tex define lemma leqTransitivity  
   as "leqTransitivity"  end define end math ]"

\item "[ math tex define lemma leqAddition  
   as "leqAddition"  end define end math ]"

\item "[ math tex define lemma reciprocal  
   as "Reciprocal"  end define end math ]"

\item "[ math tex define lemma equality  
   as "Equality"  end define end math ]"

\item "[ math tex define lemma eqLeq  
   as "eqLeq"  end define end math ]"

\item "[ math tex define lemma eqAddition  
   as "eqAddition"  end define end math ]"

\item "[ math tex define lemma eqMultiplication  
   as "eqMultiplication"  end define end math ]"

\item "[ math tex define lemma eqReflexivity  ""; EQUALITY
   as "eqReflexivity"  end define end math ]"

\item "[ math tex define lemma eqSymmetry  
   as "eqSymmetry"  end define end math ]"

\item "[ math tex define lemma eqTransitivity  
   as "eqTransitivity"  end define end math ]" 

\item "[ math tex define lemma eqTransitivity4  
   as "eqTransitivity4"  end define end math ]"

\item "[ math tex define lemma eqTransitivity5  
   as "eqTransitivity5"  end define end math ]"

\item "[ math tex define lemma eqTransitivity6  
   as "eqTransitivity6"  end define end math ]"

\item "[ math tex define lemma plus0Left  
   as "plus0Left"  end define end math ]" 

\item "[ math tex define lemma times1Left  
   as "times1Left"  end define end math ]" 

\item "[ math tex define lemma eqMultiplicationLeft  
   as "EqMultiplicationLeft"  end define end math ]" 

\item "[ math tex define lemma distributionLeft
   as "DistributionLeft"  end define end math ]" 

\item "[ math tex define lemma distributionOut
   as "DistributionOut"  end define end math ]" 

\item "[ math tex define lemma distributionOutLeft
   as "DistributionOutLeft"  end define end math ]"

\item "[ math tex define lemma three2twoTerms  
   as "Three2twoTerms"  end define end math ]" 

\item "[ math tex define lemma three2threeTerms  
   as "Three2threeTerms"  end define end math ]"

\item "[ math tex define lemma three2twoFactors  
   as "Three2twoFactors"  end define end math ]" 

\item "[ math tex define lemma three2threeFactors  
   as "Three2threeFactors"  end define end math ]"

\item "[ math tex define lemma addEquations  
   as "AddEquations"  end define end math ]" 

\item "[ math tex define lemma subtractEquations  
   as "SubtractEquations"  end define end math ]" 

\item "[ math tex define lemma subtractEquationsLeft  
   as "SubtractEquationsLeft"  end define end math ]" 

\item "[ math tex define lemma multiplyEquations  
   as "MultiplyEquations"  end define end math ]" 

\item "[ math tex define lemma eqNegated  
   as "EqNegated"  end define end math ]" 

\item "[ math tex define lemma positiveToRight(Eq)  

   as "PositiveToRight(Eq)"  end define end math ]" 

\item "[ math tex define lemma positiveToLeft(Eq)
   as "PositiveToLeft(Eq)"  end define end math ]"

\item "[ math tex define lemma positiveToLeft(Eq)(1 term)  
   as "PositiveToLeft(Eq)(1 term)"  end define end math ]" 

\item "[ math tex define lemma negativeToLeft(Eq)  
   as "NegativeToLeft(Eq)"  end define end math ]"

\item "[ math tex define lemma reciprocalToLeft(Less)  
   as "reciprocalToLeft(Less)"  end define end math ]" 

\item "[ math tex define lemma lessNeq  ""; NO EQUALITY
   as "LessNeq"  end define end math ]" 

\item "[ math tex define lemma neqSymmetry  
   as "NeqSymmetry"  end define end math ]" 

\item "[ math tex define lemma neqNegated   
   as "NeqNegated"  end define end math ]" 

\item "[ math tex define lemma subNeqRight  
   as "SubNeqRight"  end define end math ]" 

\item "[ math tex define lemma subNeqLeft  
   as "SubNeqLeft"  end define end math ]" 

\item "[ math tex define lemma negativeToRight(Neq)(1 term)  
   as "NegativeToRight(Neq)(1 term)"  end define end math ]" 

\item "[ math tex define lemma neqAddition  
   as "NeqAddition"  end define end math ]" 

\item "[ math tex define lemma neqMultiplication  
   as "NeqMultiplication"  end define end math ]" 

\item "[ math tex define lemma uniqueNegative  ""; NEGATIVE
   as "UniqueNegative"  end define end math ]" 

\item "[ math tex define lemma doubleMinus  
   as "DoubleMinus"  end define end math ]" 

\item "[ math tex define lemma leqMultiplicationLeft ""; LEQ
   as "LeqMultiplicationLeft "  end define end math ]" 

\item "[ math tex define lemma leqLessEq  
   as "LeqLessEq"  end define end math ]" 

\item "[ math tex define lemma lessLeq 
   as "LessLeq"  end define end math ]" 

\item "[ math tex define lemma from leqGeq  
   as "FromLeqGeq"  end define end math ]" 

\item "[ math tex define lemma subLeqRight  
   as "subLeqRight"  end define end math ]" 

\item "[ math tex define lemma subLeqLeft  
   as "subLeqLeft"  end define end math ]"

\item "[ math tex define lemma leqPlus1  
   as "Leq+1"  end define end math ]" 

\item "[ math tex define lemma positiveToRight(Leq)  
   as "PositiveToRight(Leq)"  end define end math ]" 

\item "[ math tex define lemma positiveToRight(Leq)(1 term)  
   as "PositiveToRight(Leq)(1 term)"  end define end math ]"

\item "[ math tex define lemma positiveToLeft(Leq)  
   as "PositiveToLeft(Leq)"  end define end math ]"

\item "[ math tex define lemma leqAdditionLeft  
   as "LeqAdditionLeft"  end define end math ]" 

\item "[ math tex define lemma leqSubtraction  
   as "leqSubtraction"  end define end math ]"

\item "[ math tex define lemma leqSubtractionLeft  
   as "leqSubtractionLeft"  end define end math ]"

\item "[ math tex define lemma leqMultiplication  
   as "leqMultiplication"  end define end math ]"

\item "[ math tex define lemma thirdGeq  
   as "thirdGeq"  end define end math ]" 

\item "[ math tex define lemma leqNegated  
   as "LeqNegated" end define end math ]" 

\item "[ math tex define lemma addEquations(Leq)  
   as "AddEquations(Leq)" end define end math ]" 

\item "[ math tex define lemma multiplyEquations(Leq)  
   as "MultiplyEquations(Leq)" end define end math ]"

\item "[ math tex define lemma leqNeqLess ""; LESS  
   as "LeqNeqLess"  end define end math ]" 

\item "[ math tex define lemma fromLess  
   as "FromLess"  end define end math ]" 

\item "[ math tex define lemma toLess  
   as "ToLess"  end define end math ]"

\item "[ math tex define lemma fromNotLess  
   as "fromNotLess"  end define end math ]"

\item "[ math tex define lemma toNotLess  
   as "toNotLess"  end define end math ]" 

\item "[ math tex define lemma lessAddition  
   as "LessAddition"  end define end math ]" 

\item "[ math tex define lemma lessAdditionLeft  
   as "LessAdditionLeft"  end define end math ]"

\item "[ math tex define lemma lessMultiplication  
   as "LessMultiplication"  end define end math ]" 


\item "[ math tex define lemma lessMultiplicationLeft  
   as "LessMultiplicationLeft"  end define end math ]"

\item "[ math tex define lemma lessDivision  
   as "LessDivision"  end define end math ]" 

\item "[ math tex define lemma positiveToRight(Less) 
   as "PositiveToRight(Less)"  end define end math ]" 

\item "[ math tex define lemma positiveToLeft(Less) 
   as "PositiveToLeft(Less)"  end define end math ]" 

\item "[ math tex define lemma negativeToLeft(Less) 
   as "NegativeToLeft(Less)"  end define end math ]" 

\item "[ math tex define lemma negativeToRight(Less) 
   as "NegativeToRight(Less)"  end define end math ]" 

\item "[ math tex define lemma addEquations(Less)  
   as "AddEquations(Less)"  end define end math ]" 

\item "[ math tex define lemma addEquations(LeqLess)  
   as "AddEquations(LeqLess)"  end define end math ]"

\item "[ math tex define lemma negativeLessPositive 
   as "NegativeLessPositive"  end define end math ]" 

\item "[ math tex define lemma leqLessTransitivity  
   as "leqLessTransitivity"  end define end math ]" 

\item "[ math tex define lemma lessLeqTransitivity  
   as "LessLeqTransitivity"  end define end math ]" 

\item "[ math tex define lemma lessTransitivity  
   as "LessTransitivity"  end define end math ]" 

\item "[ math tex define lemma lessTotality  
   as "LessTotality"  end define end math ]" 

\item "[ math tex define lemma subLessRight  
   as "SubLessRight"  end define end math ]" 

\item "[ math tex define lemma subLessLeft  
   as "SubLessLeft"  end define end math ]"

\item "[ math tex define lemma switchTerms(x<y-z)  
   as "SwitchTerms(x<y-z)"  end define end math ]"

\item "[ math tex define lemma switchTerms(x-y<z)  
   as "SwitchTerms(x-y<z)"  end define end math ]" 

\item "[ math tex define lemma lessNegated  
   as "LessNegated"  end define end math ]" 

\item "[ math tex define lemma positiveNonzero  
   as "PositiveNonzero"  end define end math ]" 

\item "[ math tex define lemma positiveNegated  
   as "PositiveNegated"  end define end math ]" 

\item "[ math tex define lemma nonpositiveNegated  
   as "NonpositiveNegated"  end define end math ]"

\item "[ math tex define lemma negativeNegated  
   as "NegativeNegated"  end define end math ]" 

\item "[ math tex define lemma nonnegativeNegated  
   as "NonnegativeNegated"  end define end math ]" 

\item "[ math tex define lemma positiveInverted  
   as "PositiveInverted"  end define end math ]" 

\item "[ math tex define lemma positiveHalved  
   as "PositiveHalved"  end define end math ]" 

\item "[ math tex define lemma nonnegativeNumerical  ""; NUMERISK
   as "NonnegativeNumerical"  end define end math ]" 

\item "[ math tex define lemma negativeNumerical  
   as "NegativeNumerical"  end define end math ]" 

\item "[ math tex define lemma positiveNumerical  
   as "PositiveNumerical"  end define end math ]" 

\item "[ math tex define lemma |0|=0  
   as "|0|=0"  end define end math ]" 

\item "[ math tex define lemma 0<=|x|  
   as "0<=|x|"  end define end math ]" 

\item "[ math tex define lemma x<=|x|  
   as "x<=|x|"  end define end math ]"

\item "[ math tex define lemma fromPositiveNumerical 
   as "FromPositiveNumerical"  end define end math ]" 

\item "[ math tex define lemma sameNumerical 
   as "SameNumerical" end define end math ]" 

\item "[ math tex define lemma signNumerical(+)  
   as "SignNumerical(+)" end define end math ]" 

\item "[ math tex define lemma signNumerical
   as "SignNumerical" end define end math ]"

\item "[ math tex define lemma toNumericalLess  
   as "ToNumericalLess"  end define end math ]" 

\item "[ math tex define lemma fromNumericalGreater  
   as "FromNumericalGreater"  end define end math ]" 

\item "[ math tex define lemma numericalDifference  
   as "NumericalDifference"  end define end math ]" 

\item "[ math tex define lemma numericalDifferenceLess helper  
   as "NumericalDifferenceLess(Helper)"  end define end math ]"

\item "[ math tex define lemma numericalDifferenceLess 
   as "NumericalDifferenceLess"  end define end math ]"

\item "[ math tex define lemma splitNumericalSumHelper  
   as "SplitNumericalSumHelper"  end define end math ]" 

\item "[ math tex define lemma splitNumericalSum(++)  
   as "splitNumericalSum(++)"  end define end math ]" 

\item "[ math tex define lemma splitNumericalSum(--)  
   as "splitNumericalSum(--)"  end define end math ]" 

\item "[ math tex define lemma splitNumericalSum(+-, smallNegative)  
   as "splitNumericalSum(+-small)"  end define end math ]" 

\item "[ math tex define lemma splitNumericalSum(+-, bigNegative)  
   as "splitNumericalSum(+-big)"  end define end math ]" 

\item "[ math tex define lemma splitNumericalSum(+-)  
   as "splitNumericalSum(+-)"  end define end math ]" 

\item "[ math tex define lemma splitNumericalSum(-+)  
   as "splitNumericalSum(-+)"  end define end math ]" 

\item "[ math tex define lemma splitNumericalSum 
   as "splitNumericalSum"  end define end math ]"

\item "[ math tex define lemma splitNumericalProduct(++)
   as "SplitNumericalProduct(++)"  end define end math ]"


\item "[ math tex define lemma splitNumericalProduct(+-)
   as "SplitNumericalProduct(+-)"  end define end math ]"

\item "[ math tex define lemma splitNumericalProduct

   as "SplitNumericalProduct"  end define end math ]"

\item "[ math tex define lemma insertMiddleTerm(Numerical)  
   as "insertMiddleTerm(Numerical)"  end define end math ]" 

\item "[ math tex define lemma insertTwoMiddleTerms(Numerical)  
   as "insertTwoMiddleTerms(Numerical)"  end define end math ]" 

\item "[ math tex define lemma leqMax1  
   as "MaxLeq(1)"  end define end math ]" 

\item "[ math tex define lemma leqMax2  
   as "MaxLeq(2)"  end define end math ]" 

\item "[ math tex define lemma lessThanMax  
   as "LessThanMax"  end define end math ]" 

\item "[ math tex define lemma x+y=zBackwards  ""; REGNESTYKKER
   as "x+y=zBackwards"  end define end math ]" 

\item "[ math tex define lemma x*y=zBackwards 
   as "x*y=zBackwards"  end define end math ]" 

\item "[ math tex define lemma x=x+(y-y) 
   as "x=x+(y-y)"  end define end math ]" 

\item "[ math tex define lemma x=x+y-y  
   as "x=x+y-y"  end define end math ]" 

\item "[ math tex define lemma x=x*y*(1/y)  
   as "x=x*y*(1/y)"  end define end math ]" 

\item "[ math tex define lemma insertMiddleTerm(Sum)  
   as "insertMiddleTerm(Sum)"  end define end math ]" 

\item "[ math tex define lemma insertTwoMiddleTerms(Sum)  
   as "insertTwoMiddleTerms(Sum)"  end define end math ]" 

\item "[ math tex define lemma insertMiddleTerm(Difference)  
   as "insertMiddleTerm(Difference)"  end define end math ]"

\item "[ math tex define lemma x*0+x=x  
   as "x*0+x=x"  end define end math ]" 

\item "[ math tex define lemma nonnegativeFactors  
   as "NonnegativeFactors"  end define end math ]" 

\item "[ math tex define lemma nonzeroFactors  
   as "NonzeroFactors"  end define end math ]"

\item "[ math tex define lemma positiveFactors  
   as "PositiveFactors"  end define end math ]"

\item "[ math tex define lemma plusTimesMinus  
   as "PlusTimesMinus"  end define end math ]" 

\item "[ math tex define lemma minusTimesMinus  
   as "MinusTimesMinus"  end define end math ]"

\item "[ math tex define lemma x*0=0  
   as "x*0=0"  end define end math ]" 

\item "[ math tex define lemma (-1)*(-1)+(-1)*1=0  
   as "(-1)*(-1)+(-1)*1=0"  end define end math ]" 

\item "[ math tex define lemma (-1)*(-1)=1  
   as "(-1)*(-1)=1"  end define end math ]" 

\item "[ math tex define lemma 0<1Helper  
   as "0<1Helper"  end define end math ]" 

\item "[ math tex define lemma 0<1  
   as "0<1"  end define end math ]"

\item "[ math tex define lemma 0<2  
   as "0<2"  end define end math ]"

\item "[ math tex define lemma 0<3  
   as "0<3"  end define end math ]"

\item "[ math tex define lemma 0<1/2  
   as "0<1/2"  end define end math ]"

\item "[ math tex define lemma 0<1/3  

   as "0<1/3"  end define end math ]"

\item "[ math tex define lemma x+x=2*x  
   as "TwoWholes"  end define end math ]" 

\item "[ math tex define lemma x+x+x=3*x  
   as "ThreeWholes"  end define end math ]"

\item "[ math tex define lemma (1/2)x+(1/2)x=x  
   as "TwoHalves" end define end math ]" 

\item "[ math tex define lemma (1/3)x+(1/3)x+(1/3)x=x  
   as "ThreeThirds" end define end math ]"

\item "[ math tex define lemma -x-y=-(x+y)  
   as "-x-y=-(x+y)"  end define end math ]" 

\item "[ math tex define lemma -x*y=-(x*y)  
   as "-x*y=-(x*y)"  end define end math ]" 

\item "[ math tex define lemma minusNegated  
   as "MinusNegated"  end define end math ]" 

\item "[ math tex define lemma times(-1)  
   as "Times(-1)"  end define end math ]" 

\item "[ math tex define lemma times(-1)Left  
   as "Times(-1)Left"  end define end math ]" 

\item "[ math tex define lemma -0=0  
   as "-0=0"  end define end math ]" 

\item "[ math tex define pred lemma allNegated(Imply) ""; PREDICATE LOGIC 
   as "AllNegated(Imply)"  end define end math ]" 

\item "[ math tex define pred lemma existNegated(Imply)  
   as "ExistNegated(Imply)"  end define end math ]" 

\item "[ math tex define pred lemma intro exist helper  
   as "IntroExist(Helper)"  end define end math ]" 

\item "[ math tex define pred lemma intro exist
   as "IntroExist"  end define end math ]" 

\item "[ math tex define pred lemma exist mp  
   as "ExistMP"  end define end math ]" 

\item "[ math tex define pred lemma exist mp2  
   as "ExistMP2"  end define end math ]" 

\item "[ math tex define pred lemma 2exist mp  
   as "TwiceExistMP"  end define end math ]" 

\item "[ math tex define pred lemma 2exist mp2  
   as "TwiceExistMP2"  end define end math ]" 

\item "[ math tex define pred lemma EAE mp  
   as "EAE-MP"  end define end math ]" 

\item "[ math tex define pred lemma addAll  
   as "AddAll "  end define end math ]" 

\item "[ math tex define pred lemma addExist helper1  
   as "AddExist(Helper1)"  end define end math ]" 

\item "[ math tex define pred lemma addExist helper2
   as "AddExist(Helper2)"  end define end math ]" 

\item "[ math tex define pred lemma addExist  
   as "AddExist"  end define end math ]" 

\item "[ math tex define pred lemma addExist(SimpleAnt)  
   as "AddExist(SimpleAnt)"  end define end math ]" 

\item "[ math tex define pred lemma addExist(Simple)  
   as "AddExist(Simple)"  end define end math ]"

\item "[ math tex define pred lemma addEAE  
   as "AddEAE"  end define end math ]" 

\item "[ math tex define pred lemma AEAnegated  
   as "AEA-negated"  end define end math ]" 

\item "[ math tex define pred lemma EEAnegated  
   as "EEA-negated"  end define end math ]" 


\item "[ math tex define prop lemma to negated and  ""; USORTERET
   as "ToNegatedAnd"  end define end math ]" 

\item "[ math tex define lemma positiveToRight(Less)(1 term)  
   as "PositiveToRight(Less)(1 term)"  end define end math ]" 

\item "[ math tex define pred lemma (A~)to(~E)
   as "(A~)to(~E)"  end define end math ]" 



\item "[ math tex define lemma eqTransitivity4  
   as "eqTransitivity4"  end define end math ]" 

\item "[ math tex define lemma plus0Left(R)  
   as "Plus0Left(R)"  end define end math ]" 

\item "[ math tex define lemma x=x+(y-y)(R)  
   as "x=x+(y-y)(R)"  end define end math ]" 

\item "[ math tex define lemma x=x+y-y(R)  
   as "x=x+y-y(R)"  end define end math ]" 

\item "[ math tex define lemma positiveToRight(Eq)(R)  
   as "PositiveToRight(Eq)(R)"  end define end math ]" 

\item "[ math tex define lemma subtractEquations(R)  
   as "SubtractEquations(R)"  end define end math ]" 

\item "[ math tex define lemma neqAddition(R)  
   as "NeqAddition(R)"  end define end math ]" 

\item "[ math tex define lemma eqAdditionLeft(R)  
   as "EqAdditionLeft(R)"  end define end math ]" 

\item "[ math tex define lemma three2twoTerms(R)  
   as "Three2twoTerms(R)"  end define end math ]" 

\item "[ math tex define lemma positiveToRight(Less)(R)  
   as "PositiveToRight(Less)(R)"  end define end math ]" 

\item "[ math tex define lemma three2threeTerms(R)  
   as "Three2threeTerms(R)"  end define end math ]" 

\item "[ math tex define lemma positiveToRight(Less)(1 term)(R)  
   as "PositiveToRight(Less)(1 term)(R)"  end define end math ]"

\item "[ math tex define lemma toLeq(Advanced)(R)   
   as "ToLeq(Advanced)(R)"  end define end math ]" 

\item "[ math tex define lemma leqNeqLess(R)  
   as "LeqNeqLess(R)"  end define end math ]" 

\item "[ math tex define lemma subLeqLeft(R)  
   as "SubLeqLeft(R)"  end define end math ]" 


\item "[ math tex define lemma leqLessTransitivity(R)  
   as "LeqLessTransitivity(R)"  end define end math ]" 

\item "[ math tex define lemma negativeToLeft(Eq)(R)  
   as "NegativeToLeft(Eq)(R)"  end define end math ]" 

\item "[ math tex define lemma negativeToRight(Less)(R)  
   as "NegativeToRight(Less)(R)"  end define end math ]" 

\item "[ math tex define lemma !!==Symmetry  
   as "!!==Symmetry" end define end math ]" 

\item "[ math tex define lemma negativeToRight(Eq)(R) 
   as "NegativeToRight(Eq)(R)"  end define end math ]" 

\item "[ math tex define lemma negativeToRight(Eq)(1 term)(R)  
   as "NegativeToRight(Eq)(1 term)(R)"  end define end math ]" 


\item "[ math tex define lemma subLeqRight(R)  
   as "SubLeqRight(R)"  end define end math ]" 

\item "[ math tex define lemma doubleMinus(R)  
   as "DoubleMinus(R)"  end define end math ]" 

\item "[ math tex define lemma uniqueNegative(R)  
   as "UniqueNegative(R)"  end define end math ]" 

\item "[ math tex define lemma subtractEquationsLeft(R)  
   as "SubtractEquationsLeft(R)"  end define end math ]" 

\item "[ math tex define lemma eqNegated(R)  
   as "EqNegated(R)"  end define end math ]" 

\item "[ math tex define lemma neqNegated(R)  
   as "NeqNegated(R)"  end define end math ]" 

\item "[ math tex define lemma leqNegated(R)  
   as "LeqNegated(R)"  end define end math ]" 

\item "[ math tex define lemma lessNegated(R)
   as "LessNegated(R)"  end define end math ]" 

\item "[ math tex define lemma -0=0(R)  
   as "-0=0(R)"  end define end math ]" 

\item "[ math tex define lemma negativeNegated(R)
   as "NegativeNegated(R)"  end define end math ]" 

\item "[ math tex define lemma from leqGeq(R)  
   as "FromLeqGeq(R)"  end define end math ]" 

\item "[ math tex define lemma fromLess(R)  
   as "FromLess(R)"  end define end math ]" 

\item "[ math tex define lemma nonnegativeNumerical(R)
   as "NonnegativeNumerical(R)"  end define end math ]" 

\item "[ math tex define lemma negativeNumerical(R)  
   as "NegativeNumerical(R)"  end define end math ]" 

\item "[ math tex define lemma 0<=|x|(R)  
   as "0<=|x|(R)"  end define end math ]" 

\item "[ math tex define lemma positiveNegated(R)  
   as "PositiveNegated(R)"  end define end math ]" 

\item "[ math tex define lemma addEquations(R)  
   as "AddEquations(R)"  end define end math ]" 

\item "[ math tex define lemma distributionOut(R)  
   as "DistributionOut(R)"  end define end math ]" 

\item "[ math tex define lemma ==Transitivity5  
   as "==Transitivity5"  end define end math ]" 


\item "[ math tex define lemma x*0=0(R)fff  
   as "x*0=0(R)(fff)"  end define end math ]"

\item "[ math tex define lemma x*0+x=x(R)  
   as "x*0+x=x(R)"  end define end math ]" 

\item "[ math tex define lemma times(-1)(R)  
   as "Times(-1)(R)"  end define end math ]" 
 
\item "[ math tex define lemma times(-1)Left(R)  
   as "Times(-1)Left(R)"  end define end math ]" 

\item "[ math tex define lemma -x-y=-(x+y)(R)  
   as "-x-y=-(x+y)(R)"  end define end math ]" 

\item "[ math tex define lemma lessTotality(R)  
   as "LessTotality(R)"  end define end math ]" 
 
\item "[ math tex define lemma signNumerical(+)(R)
   as "SignNumerical(+)(R)"  end define end math ]" 

\item "[ math tex define lemma positiveNumerical(R)  
   as "PositiveNumerical(R)"  end define end math ]" 

\item "[ math tex define lemma sameNumerical(R)  
   as "SameNumerical(R)"  end define end math ]" 

\item "[ math tex define lemma minusNegated(R)  
   as "MinusNegated(R)"  end define end math ]" 

\item "[ math tex define lemma signNumerical(R)  
   as "SignNumerical(R)"  end define end math ]" 

\item "[ math tex define lemma numericalDifference(R)  
   as "NumericalDifference(R)"  end define end math ]" 

\item "[ math tex define lemma USlimitIsUpperBound helper  
   as "USlimitIsUpperBound(Helper)"  end define end math ]" 

\item "[ math tex define lemma USlimitIsUpperBound  
   as "USlimitIsUpperBound"  end define end math ]"

\item "[ math tex define lemma x<=|x|(R)  
   as "x<=|x|(R)"  end define end math ]" 

\item "[ math tex define lemma (-1)*(-1)+(-1)*1=0(R)  
   as "(-1)*(-1)+(-1)*1=0(R)"  end define end math ]" 

\item "[ math tex define lemma (-1)*(-1)=1(R) 
   as "(-1)*(-1)=1(R)"  end define end math ]" 

\item "[ math tex define lemma 0<1Helper(R)  
   as "0<1Helper(R)"  end define end math ]" 

\item "[ math tex define lemma 0<1(R)  
   as "0<1(R)"  end define end math ]" 

\item "[ math tex define lemma expZero exact(R) 
   as "ExpZero(Exact)(R)"  end define end math ]" 

\item "[ math tex define lemma positiveBase(R) base 
   as "PositiveBase(R)(Base)"  end define end math ]" 

\item "[ math tex define lemma three2twoFactors(R) 
   as "Three2twoFactors(R)"  end define end math ]" 

\item "[ math tex define lemma x=x*y*(1/y)(R)  
   as "x=x*y*(1/y)(R)"  end define end math ]" 

\item "[ math tex define lemma neqMultiplication(R) 
   as "NeqMultiplication(R)"  end define end math ]" 

\item "[ math tex define lemma lessTransitivity(R)  
   as "LessTransitivity(R)"  end define end math ]"

\item "[ math tex define lemma 0<2(R) 
   as "0<2(R)"  end define end math ]" 

\item "[ math tex define lemma sameExp(R) base 
   as "SameExp(R)(Base)"  end define end math ]" 

\item "[ math tex define lemma sameExp(R) indu 
   as "SameExp(R)(Indu)"  end define end math ]" 

\item "[ math tex define lemma sameExp(R) 
   as "SameExp(R)"  end define end math ]"

\item "[ math tex define lemma subNeqLeft(R) 
   as "SubNeqLeft(R)"  end define end math ]" 

\item "[ math tex define lemma subNeqRight(R) 
   as "SubNeqRight(R)"  end define end math ]" 

\item "[ math tex define lemma nonzeroFactors(R) 
   as "NonzeroFactors(R)"  end define end math ]" 

\item "[ math tex define lemma nonnegativeFactors(R) 
   as "NonnegativeFactors(R)"  end define end math ]" 

\item "[ math tex define lemma positiveFactors(R)  
   as "PositiveFactors(R)"  end define end math ]" 

\item "[ math tex define lemma lessDivision(R) 
   as "LessDivision(R)"  end define end math ]" 

\item "[ math tex define lemma 0<1/2(R) 
   as "0<1/2(R)"  end define end math ]" 

\item "[ math tex define lemma positiveToRight(Eq)(1 term)(R) 
   as "PositiveToRight(Eq)(1 term)(R)"  end define end math ]" 

\item "[ math tex define lemma exp(+1)(R) 
   as "Exp(+1)(R)"  end define end math ]" 

\item "[ math tex define lemma positiveBase(R) indu 
   as "PositiveBase(R)(Indu)"  end define end math ]" 

\item "[ math tex define lemma positiveBase(R) 
   as "PositiveBase(R)"  end define end math ]"

\item "[ math tex define  lemma -x*y=-(x*y)(R) 
   as "-x*y=-(x*y)(R)"  end define end math ]" 

\item "[ math tex define lemma positiveToLeft(Eq)(R)  
   as "PositiveToLeft(Eq)(R)"  end define end math ]" 

\item "[ math tex define lemma times1Left(R)  
   as "Times1Left(R)"  end define end math ]" 

\item "[ math tex define lemma ==Transitivity6 
   as "==Transitivity6"  end define end math ]" 

\item "[ math tex define lemma x+x=2*x(R) 
   as "x+x=2*x(R)"  end define end math ]" 

\item "[ math tex define lemma (1/2)x+(1/2)x=x(R) 
   as "(1/2)x+(1/2)x=x(R)"  end define end math ]" 

\item "[ math tex define lemma distributionOut(Minus)(R) 
   as "DistributionOut(Minus)(R)"  end define end math ]" 

\item "[ math tex define lemma (1/2)(x+y)-x=(1/2)(y-x)(R) 

   as "(1/2)(x+y)-x=(1/2)(y-x)(R)" end define end math ]" 

\item "[ math tex define lemma intervalSize(R) base  
   as "IntervalSize(R)(Base)"  end define end math ]"

\item "[ math tex define lemma lessMultiplicationLeft(R) 
   as "LessMultiplicationLeft(R)"  end define end math ]" 

\item "[ math tex define lemma negativeToLeft(Less)(R) 
   as "NegativeToLeft(Less)(R)"  end define end math ]" 

\item "[ math tex define lemma negativeToLeft(Less)(1 term)(R)  
   as "NegativeToLeft(Less)(1 term)(R)"  end define end math ]" 

\item "[ math tex define lemma y-(1/2)(x+y)=(1/2)(y-x)(R)  
   as "y-(1/2)(x+y)=(1/2)(y-x)(R)"  end define end math ]" 

\item "[ math tex define lemma intervalSize(R) indu  
   as "IntervalSize(R)(Indu)"  end define end math ]" 

\item "[ math tex define lemma intervalSize(R)  
   as "IntervalSize(R)"  end define end math ]" 

\item "[ math tex define lemma XSlessUS(R)  
   as "XSlessUS(R)"  end define end math ]" 

\item "[ math tex define lemma USdecreasing(+1)(R)  
   as "USdecreasing(+1)(R)"  end define end math ]" 

\item "[ math tex define lemma expUnbounded base 
   as "ExpUnbounded(Base)"  end define end math ]" 

\item "[ math tex define lemma expUnbounded indu 
   as "ExpUnbounded(Indu)"  end define end math ]" 

\item "[ math tex define lemma expUnbounded 
   as "ExpUnbounded"  end define end math ]" 
 
\item "[ math tex define lemma 1<=x+1(N) 
   as "1<=x+1(N)"  end define end math ]" 

\item "[ math tex define lemma nonzeroProduct(2)(R) 
   as "NonzeroProduct(2)(R)"  end define end math ]" 

\item "[ math tex define lemma positiveNonzero(R) 
   as "PositiveNonzero(R)"  end define end math ]" 

\item "[ math tex define lemma nonreciprocalToRight(Eq)(1 term)(R) 
   as "NonreciprocalToRight(Eq)(1 term)(R)"  end define end math ]" 

\item "[ math tex define lemma expNonzero base 
   as "ExpNonzero(Base)"  end define end math ]" 

\item "[ math tex define lemma expNonzero indu 
   as "ExpNonzero(Indu)"  end define end math ]" 

\item "[ math tex define lemma expNonzero 
   as "ExpNonzero"  end define end math ]"

\item "[ math tex define lemma expNonzero(2) 
   as "ExpNonzero(2)"  end define end math ]" 

\item "[ math tex define lemma multiplyEquations(R) 
   as "MultiplyEquations(R)"  end define end math ]" 

\item "[ math tex define lemma halfBase base 
   as "HalfBase(Base)"  end define end math ]" 

\item "[ math tex define lemma halfBase indu 
   as "HalfBase(Indu)"  end define end math ]" 

\item "[ math tex define lemma halfBase  
   as "HalfBase"  end define end math ]" 


\item "[ math tex define lemma three2threeFactors(R)  
   as "Three2threeFactors(R)"  end define end math ]"

\item "[ math tex define lemma x*y=zBackwards(R) 
   as "x*y=zBackwards(R)"  end define end math ]" 

\item "[ math tex define lemma positiveInverted(R) 
   as "PositiveInverted(R)"  end define end math ]" 

\item "[ math tex define lemma reciprocalToRight(Less)(R) 
   as "ReciprocalToRight(Less)(R)"  end define end math ]" 

\item "[ math tex define lemma reciprocalToRight(Less)(1 term)(R) 
   as "ReciprocalToRight(Less)(1 term)(R)"  end define end math ]" 

\item "[ math tex define lemma nonreciprocalToLeft(Less)(R) 
   as "NonreciprocalToLeft(Less)(R)"  end define end math ]" 

\item "[ math tex define lemma 1<x*y(R) 
   as "1<x*y(R)"  end define end math ]" 

\item "[ math tex define lemma switchFactors(1/x<y)(R)  
   as "SwitchFactors(1/x<y)(R)"  end define end math ]" 

\item "[ math tex define lemma smallHalving 
   as "SmallHalving"  end define end math ]" 

\item "[ math tex define lemma intervalSize(anyPositive) 
   as "IntervalSize(anyPositive)" end define end math ]" 

\item "[ math tex define lemma USdecreasing(+n) base 
   as "USdecreasing(+n)(Base)"  end define end math ]" 

\item "[ math tex define lemma USdecreasing(+n) indu 
   as "USdecreasing(+n)(Indu)"  end define end math ]" 

\item "[ math tex define lemma USdecreasing(+n) 
   as "USdecreasing(+n)"  end define end math ]" 

\item "[ math tex define lemma USdecreasing 
   as "USdecreasing"  end define end math ]"

\item "[ math tex define lemma leqAdditionLeft(R)  
   as "LeqAdditionLeft(R)"  end define end math ]" 

\item "[ math tex define lemma toNotLess(R)  
   as "ToNotLess(R)"  end define end math ]" 

\item "[ math tex define lemma limitOfUSIsLeq
   as "LimitOfUSIsLeq"  end define end math ]"

\item "[ math tex define lemma subtractEquations(Less)(R) 
   as "SubtractEquations(Less)(R)"  end define end math ]" 

\item "[ math tex define lemma subtractEquationsLeft(Less)(R) 
   as "SubtractEquationsLeft(Less)(R)"  end define end math ]" 

\item "[ math tex define lemma lessNegated(Negative)(R) 
   as "LessNegated(Negative)(R)"  end define end math ]" 

\item "[ math tex define prop lemma from negated and (imply) 
   as "FromNegatedAnd(Imply)"  end define end math ]" 

\item "[ math tex define prop lemma remove double neg (consequent) 
   as "RemoveDoubleNeg(Consequent)" end define end math ]" 

\item "[ math tex define lemma fromNotUpperBound 
   as "FromNotUpperBound"  end define end math ]" 

\item "[ math tex define lemma leqNUB 
   as "LeqNUB"  end define end math ]" 

\item "[ math tex define lemma USlimitIsLeastUpperBound helper 
   as "USlimitIsLeastUpperBound(Helper)"  end define end math ]" 

\item "[ math tex define lemma USlimitIsLeastUpperBound 
   as "USlimitIsLeastUpperBound"  end define end math ]"


\item "[ math tex define pred lemma exist mp3  
   as "ExistMP3"  end define end math ]" 

\item "[ math tex define lemma greaterPositive(N)  
   as "GreaterPositive(N)"  end define end math ]" 

\item "[ math tex define lemma ysFClose helper  
   as "ysFClose(Helper)"  end define end math ]" 

\item "[ math tex define lemma ysFClose
   as "ysFClose"  end define end math ]" 

\item "[ math tex define lemma ysFCauchy helper  
   as "ysFCauchy(Helper)"  end define end math ]" 

\item "[ math tex define lemma ysFCauchy
   as "ysFCauchy"  end define end math ]"

\item "[ math tex define lemma sameFsymmetry  ""; f-afdelingen
   as "SFsymmetry"  end define end math ]"

\item "[ math tex define lemma sameFtransitivity  
   as "SFtransitivity"  end define end math ]" 

\item "[ math tex define lemma =f to sameF  
   as "=fToSameF "  end define end math ]" 

\item "[ math tex define lemma plusF(Sym)  
   as "PlusF(Sym)"  end define end math ]" 

\item "[ math tex define lemma timesF(Sym)  
   as "TimesF(Sym)"  end define end math ]"

\item "[ math tex define lemma f2R(Plus)  
   as "f2R(Plus)"  end define end math ]" 

\item "[ math tex define lemma f2R(Times)  
   as "f2R(Times)"  end define end math ]"

\item "[ math tex define lemma <<TransitivityHelper(Q)
   as "<<TransitivityHelper(Q)"  end define end math ]" 

\item "[ math tex define lemma <<Transitivity
   as "<<Transitivity"  end define end math ]" 

\item "[ math tex define lemma <<==Reflexivity ""; NUMRENE
   as "<<==Reflexivity"  end define end math ]"

\item "[ math tex define lemma <<==AntisymmetryHelper(Q)
   as "<<==AntisymmetryHelper(Q)"  end define end math ]" 

\item "[ math tex define lemma fromNotSameF(Weak)(Helper)  
   as "FromNotSameF(Weak)(Helper)"  end define end math ]" 

\item "[ math tex define lemma fromNotSameF(Weak)  
   as "FromNotSameF(Weak)"  end define end math ]"

\item "[ math tex define lemma fromNotLess(F)  
   as "FromNotLess(F)"  end define end math ]" 

\item "[ math tex define lemma plus0(F)  
   as "Plus0(F)"  end define end math ]" 
\item "[ math tex define lemma ==Addition  
   as "==Addition"  end define end math ]" 

\item "[ math tex define lemma ==AdditionLeft  
   as "==AdditionLeft"  end define end math ]" 
\item "[ math tex define lemma fpart-Bounded base  
   as "Fpart-Bounded(Base)"  end define end math ]" 

\item "[ math tex define lemma fpart-Bounded indu helper  
   as "Fpart-Bounded(InduHelper)"  end define end math ]" 

\item "[ math tex define lemma fpart-Bounded indu  
   as "Fpart-Bounded(Indu)"  end define end math ]" 

\item "[ math tex define lemma fpart-Bounded  
   as "Fpart-Bounded"  end define end math ]" 

\item "[ math tex define lemma f-Bounded  
   as "F-Bounded"  end define end math ]"

\item "[ math tex define lemma f-Bounded helper
   as "F-Bounded(Helper)"  end define end math ]" 

\item "[ math tex define lemma sameFmultiplication helper  
   as "SameFmultiplication(Helper)"  end define end math ]"

\item "[ math tex define lemma sameFmultiplication  
   as "SameFmultiplication"  end define end math ]"

\item "[ math tex define lemma fromNot<f(Weak) helper  
   as "FromNot<f(Weak)(Helper)"  end define end math ]" 

\item "[ math tex define lemma fromNot<f(Weak)  
   as "FromNot<f(Weak)"  end define end math ]"

\item "[ math tex define lemma fromNot<f(Strong) helper2  
   as "FromNot<f(Strong)(Helper2)"  end define end math ]" 

\item "[ math tex define lemma fromNot<f(Strong) helper  
   as "FromNot<f(Strong)(Helper)"  end define end math ]"

\item "[ math tex define lemma fromNot<f(Strong)  
   as "FromNot<f(Strong)"  end define end math ]"

\item "[ math tex define lemma fromNotSameF(Strongest) helper2  
   as "fromNotSameF(Strongest)(Helper2)"  end define end math ]" 

\item "[ math tex define lemma fromNotSameF(Strongest) helper  
   as "fromNotSameF(Strongest)(Helper)"  end define end math ]"

\item "[ math tex define lemma fromNotSameF(Strongest)  
   as "fromNotSameF(Strongest)"  end define end math ]"

\item "[ math tex define lemma toLess(F) helper  
   as "ToLess(F)(Helper)"  end define end math ]" 

\item "[ math tex define lemma toLess(F)  
   as "ToLess(F)"  end define end math ]"

\item "[ math tex define lemma lessMultiplication(F) helper2
   as "LessMultiplication(F)(Helper2)"  end define end math ]" 

\item "[ math tex define lemma lessMultiplication(F) helper
   as "LessMultiplication(F)(Helper)"  end define end math ]" 

\item "[ math tex define lemma lessMultiplication(F)
   as "LessMultiplication(F)"  end define end math ]"

\item "[ math tex define lemma eqMultiplication(R)  
   as "EqMultiplication(R)"  end define end math ]" 

\item "[ math tex define lemma eqMultiplicationLeft(R)  
   as "EqMultiplicationLeft(R)"  end define end math ]" 

\item "[ math tex define lemma plusAssociativity(F)  
   as "PlusAssociativity(F)"  end define end math ]" 

\item "[ math tex define lemma fromNot<<  
   as "FromNot<<"  end define end math ]" 

\item "[ math tex define lemma toLess(R)  
   as "ToLess(R)"  end define end math ]" 

\item "[ math tex define lemma leqTotality(R)  
   as "LeqTotality(R)"  end define end math ]" 

\item "[ math tex define lemma x*0=0(F)  
   as "x*0=0(F)"  end define end math ]" 

\item "[ math tex define lemma x*0=0(R)  
   as "x*0=0(R)"  end define end math ]"

\item "[ math tex define lemma plusCommutativity(F)  
   as "PlusCommutativity(F)"  end define end math ]" 

\item "[ math tex define lemma 2cauchy helper 
   as "Cauchy(2)(Helper)"  end define end math ]" 

\item "[ math tex define lemma 2cauchy 
   as "Cauchy(2)"  end define end math ]" 

\item "[ math tex define lemma timesAssociativity(F)  
   as "TimesAssociativity(F)"  end define end math ]" 

\item "[ math tex define lemma lessMultiplication(R)
   as "LessMultiplication(R)"  end define end math ]" 

\item "[ math tex define lemma leqMultiplication(R)
   as "LeqMultiplication(R)"  end define end math ]"

\item "[ math tex define lemma times1f  
   as "Times1f"  end define end math ]" 

\item "[ math tex define lemma reciprocalF nonzero  
   as "ReciprocalFnonzero"  end define end math ]" 

\item "[ math tex define lemma reciprocalFny nonzero  
   as "ReciprocalFnynonzero"  end define end math ]" 

\item "[ math tex define lemma eventually=f to sameF helper  
   as "(Eventually=f)2sameF(Helper)"  end define end math ]" 

\item "[ math tex define lemma eventually=f to sameF  
   as "(Eventually=f)2sameF"  end define end math ]" 

\item "[ math tex define lemma fromNotSameF(Strong) helper2  
   as "FromNotSameF(Strong)(Helper2)"  end define end math ]"
\item "[ math tex define lemma fromNotSameF(Strong) helper  
   as "FromNotSameF(Strong)(Helper)"  end define end math ]" 

\item "[ math tex define lemma fromNotSameF(Strong)  
   as "FromNotSameF(Strong)"  end define end math ]" 

\item "[ math tex define lemma sameFreciprocal helper  
   as "SameFreciprocal(Helper)"  end define end math ]" 

\item "[ math tex define lemma sameFreciprocal  
   as "SameFreciprocal"  end define end math ]"

\item "[ math tex define lemma from!!==  
   as "From!!=="  end define end math ]" 

\item "[ math tex define lemma reciprocal(R)  
   as "Reciprocal(R)"  end define end math ]" 

\item "[ math tex define lemma timesCommutativity(F)  
   as "TimesCommutativity(F)"  end define end math ]" 

\item "[ math tex define lemma distribution(F)  
   as "Distribution(F)"  end define end math ]" 

\item "[ math tex define lemma fromNotLess(R)  
   as "FromNotLess(R)"  end define end math ]" 

\item "[ math tex define prop lemma to negated and(1)
  as "ToNegatedAnd(1)"  end define end math ]" 

\item "[ math tex define lemma fromMax(1)  
   as "FromMax(1)"  end define end math ]" 

\item "[ math tex define lemma fromMax(2)  
   as "FromMax(2)"  end define end math ]" 


\item "[ math tex define lemma cartProdIsRelation
  as "CartProdIsRelation"  end define end math ]" 


\item "[ math tex define lemma fromSubset
  as "FromSubset"  end define end math ]" 

\item "[ math tex define lemma subsetIsRelation
  as "SubsetIsRelation"  end define end math ]" 

\item "[ math tex define lemma seriesSubsetCP  
   as "SeriesSubsetCP"  end define end math ]" 

\item "[ math tex define lemma valueType  
   as "ValueType"  end define end math ]" 

\item "[ math tex define lemma toSeries  
   as "ToSeries"  end define end math ]" 

\item "[ math tex define lemma fromSeries  
   as "FromSeries"  end define end math ]" 

\item "[ math tex define prop lemma remove or  
   as "RemoveOr"  end define end math ]" 

\item "[ math tex define lemma fromSingleton  
   as "FromSingleton"  end define end math ]" 

\item "[ math tex define lemma inPair(1)  
   as "InPair(1)"  end define end math ]" 

\item "[ math tex define lemma inPair(2)  
   as "InPair(2)"  end define end math ]" 

\item "[ math tex define lemma sameMember(2)  
   as "SameMember(2)"  end define end math ]" 

\item "[ math tex define lemma toBinaryUnion(1)  
   as "ToBinaryUnion(1)"  end define end math ]" 

\item "[ math tex define lemma toBinaryUnion(2)  
   as "ToBinaryUnion(2)"  end define end math ]" 

\item "[ math tex define lemma fromOrderedPair(twoLevels)  
   as "FromOrderedPair(TwoLevels)"  end define end math ]" 

\item "[ math tex define lemma toCartProd helper  
   as "ToCartProd(Helper)"  end define end math ]" 

\item "[ math tex define lemma toCartProd
   as "ToCartProd"  end define end math ]"

\item "[ math tex define lemma nonreciprocalToRight(Eq)  
   as "NonreciprocalToRight(Eq)"  end define end math ]" 

\item "[ math tex define lemma nonreciprocalToLeft(Eq)(1 term)  
   as "NonreciprocalToLeft(Eq)(1 term)"  end define end math ]" 

\item "[ math tex define lemma sameReciprocal
     as "SameReciprocal"  end define end math ]" 


\item "[ math tex define lemma CPseparationIsRelation  
   as "CPseparationIsRelation"  end define end math ]" 

\item "[ math tex define lemma orderedPairEquality  
   as "OrderedPairEquality"  end define end math ]" 

\item "[ math tex define lemma reciprocalIsFunction  
   as "ReciprocalIsFunction"  end define end math ]"

\item "[ math tex define lemma reciprocalIsTotal  
   as "ReciprocalIsTotal"  end define end math ]"

\item "[ math tex define lemma reciprocalIsRationalSeries
   as "ReciprocalIsRationalSeries"  end define end math ]"

\item "[ math tex define lemma crsIsRelation
  as "CrsIsRelation"  end define end math ]" 

\item "[ math tex define lemma crsIsFunction  
   as "CrsIsFunction "  end define end math ]" 

\item "[ math tex define lemma crsIsTotal  
   as "CrsIsTotal"  end define end math ]" 

\item "[ math tex define lemma crsIsSeries  
   as "CrsIsSeries"  end define end math ]" 

\item "[ math tex define lemma crsLookup  
   as "CrsLookup"  end define end math ]" 

\item "[ math tex define lemma 0f 
   as "0f"  end define end math ]"

\item "[ math tex define lemma 1f 
   as "1f"  end define end math ]" 

\item "[ math tex define lemma toSingleton  
   as "ToSingleton"  end define end math ]" 

\item "[ math tex define lemma fromSameSingleton  
   as "FromSameSingleton"  end define end math ]" 


\item "[ math tex define lemma singletonmembersEqual 
   as "SingletonmembersEqual"  end define end math ]" 

\item "[ math tex define lemma unequalsNotInSingleton  
   as "UnequalsNotInSingleton"  end define end math ]" 

\item "[ math tex define lemma nonsingletonmembersUnequal  
   as "NonsingletonmembersUnequal"  end define end math ]" 

\item "[ math tex define lemma fromOrderedPair   
   as "FromOrderedPair"  end define end math ]"

\item "[ math tex define lemma fromOrderedPair(1)   
   as "FromOrderedPair(1)"  end define end math ]"

\item "[ math tex define lemma fromOrderedPair(2)   
   as "FromOrderedPair(2)"  end define end math ]"

\item "[ math tex define lemma fromCartProd  
   as "FromCartProd"  end define end math ]" 

\item "[ math tex define lemma fromCartProd(1)  
   as "FromCartProd(1)"  end define end math ]" 

\item "[ math tex define lemma fromCartProd(2)  
   as "FromCartProd(2)"  end define end math ]" 

\item "[ math tex define lemma sameOrderedPair   
   as "sameOrderedPair"  end define end math ]" 

\item "[ math tex define lemma inSeries helper  
   as "InSeriesHelper"  end define end math ]" 

\item "[ math tex define lemma inSeries  
   as "InSeries"  end define end math ]" 

\item "[ math tex define lemma to=f subset helper 
   as "To=f(Subset)(Helper)"  end define end math ]" 

\item "[ math tex define lemma to=f subset
   as "To=f(Subset)"  end define end math ]" 

\item "[ math tex define lemma to=f
   as "To=f"  end define end math ]"

\item "[ math tex define tester1  
   as "Tester1"  end define end math ]" 

\item "[ math tex define tester2  
   as "Tester2"  end define end math ]" 

\item "[ math tex define tester3  
   as "Tester3"  end define end math ]" 

\item "[ math tex define tester4  
   as "Tester4"  end define end math ]" 

\item "[ math tex define tester5  
   as "Tester5"  end define end math ]" 

\item "[ math tex define tester6  
   as "Tester6"  end define end math ]"

\item "[ math tex define lemma productIsFunction  
   as "productIsFunction"  end define end math ]"

\item "[ math tex define lemma productIsTotal  
   as "productIsTotal"  end define end math ]"

\item "[ math tex define lemma productIsRationalSeries  
   as "ProductIsRationalSeries"  end define end math ]"

\item "[ math tex define lemma timesF  
   as "TimesF"  end define end math ]" 

\item "[ math tex define lemma -x+(1/2)x=-(1/2)x  
   as "-x+(1/2)x=-(1/2)x"  end define end math ]" 

\item "[ math tex define lemma closetolessIsLess  
   as "ClosetolessIsLess"  end define end math ]" 

\item "[ math tex define lemma subLessLeft(F)  
   as "SubLessLeft(F)"  end define end math ]"

\item "[ math tex define lemma subLessLeft(R)  
   as "SubLessLeft(R)"  end define end math ]"

\item "[ math tex define lemma closetogreaterIsGreater  
   as "ClosetogreaterIsGreater"  end define end math ]" 

\item "[ math tex define lemma subLessRight(F)  
   as "SubLessRight(F)"  end define end math ]"

\item "[ math tex define lemma subLessRight(R)  
   as "SubLessRight(R)"  end define end math ]"

\item "[ math tex define lemma positiveTripled  
   as "PositiveTripled"  end define end math ]" 

\item "[ math tex define lemma positiveDividedBy3
   as "PositiveDividedBy3"  end define end math ]" 

\item "[ math tex define lemma |x-x|=0
  as "|x-x|=0"  end define end math ]" 

\item "[ math tex define lemma 1<2  
   as "1<2"  end define end math ]" 

\item "[ math tex define lemma 1/3<2/3  
   as "1/3<2/3"  end define end math ]" 

\item "[ math tex define lemma (1/3)x+(1/3)x=(2/3)x  
   as "(1/3)x+(1/3)x=(2/3)x"  end define end math ]" 

\item "[ math tex define lemma (2/3)x+(1/3)x=x  
   as "(2/3)x+(1/3)x=x"  end define end math ]"

\item "[ math tex define lemma -x+(2/3)x=-(1/3)x
   as "-x+(2/3)x=-(1/3)x"  end define end math ]"

\item "[ math tex define lemma preserveLessGreater  
   as "PreserveLessGreater"  end define end math ]" 

\item "[ math tex define lemma -(1/3)x-(1/3)x=-(2/3)x  
   as "-(1/3)x-(1/3)x=-(2/3)x"  end define end math ]" 

\item "[ math tex define lemma -x+(1/3)x=-(2/3)x  
   as "-x+(1/3)x=-(2/3)x"  end define end math ]" 


\end{list}

\end{document}

End of file
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