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\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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 eqAdditionLeft 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 eqAdditionLeft 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 eqAddition modus ponens ell a indeed meta x + meta z = meta y + meta z 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 z = meta z + meta y end line because lemma eqTransitivity4 modus ponens ell c modus ponens ell b modus ponens ell d indeed meta z + meta x = meta z + meta y qed end math ] "
" [ math in theory system Q lemma lemma eqMultiplicationLeft 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 eqMultiplicationLeft 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 eqMultiplication modus ponens ell a indeed meta x * meta z = meta y * meta z end line line ell c because axiom timesCommutativity indeed meta z * meta x = meta x * meta z end line line ell d because axiom timesCommutativity indeed meta y * meta z = meta z * meta y end line because lemma eqTransitivity4 modus ponens ell c modus ponens ell b modus ponens ell d indeed meta z * meta x = meta z * meta y qed end math ] "
" [ math in theory system Q lemma lemma plusF(Sym) says for all terms meta m comma meta fx comma meta fy indeed [ meta fx ; meta m ] + [ meta fy ; meta m ] = [ meta fx +f meta fy ; meta m ] end lemma end math ] "
" [ math system Q proof of lemma plusF(Sym) reads any term meta m comma meta fx comma meta fy end line line ell a because axiom plusF indeed [ meta fx +f meta fy ; meta m ] = [ meta fx ; meta m ] + [ meta fy ; meta m ] end line because lemma eqSymmetry modus ponens ell a indeed [ meta fx ; meta m ] + [ meta fy ; meta m ] = [ meta fx +f meta fy ; meta m ] qed end math ] "
" [ math in theory system Q lemma lemma plus0Left says for all terms meta x indeed 0 + meta x = meta x end lemma end math ] "
" [ math system Q proof of lemma plus0Left reads any term meta x end line line ell a because axiom plus0 indeed meta x + 0 = meta x end line line ell b because axiom plusCommutativity indeed 0 + meta x = meta x + 0 end line because lemma eqTransitivity modus ponens ell b modus ponens ell a indeed 0 + meta x = meta x qed end math ] "
" [ math in theory system Q lemma lemma times1Left says for all terms meta x indeed 1 * meta x = meta x end lemma end math ] "
" [ math system Q proof of lemma times1Left 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 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 ] "
" [ math in theory system Q lemma lemma induction says for all terms meta v1 comma meta a comma meta b comma meta c indeed 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 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 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 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 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 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 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 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 ( 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 ( 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 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 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 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 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 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 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 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 (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 (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 (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 (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 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 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 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 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 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 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 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 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 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 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 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 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 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 (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 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 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 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 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 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 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 ] "
" [ 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 ] "
(*** 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 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 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 not0 parenthesis meta y <= meta x and0 not0 meta y = meta x end parenthesis end line because 1rule repetition modus ponens ell i indeed not0 meta y < meta x 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 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 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 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 ] "
(*** 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 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 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 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 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 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 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 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 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 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 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
" [ math system Q proof of lemma fromNot<< reads any term meta fx comma meta fy end line line ell a because prop lemma auto imply indeed meta fx
" [ 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 sameSeries(Gen) says for all terms meta m comma meta n comma meta fx comma meta fy comma meta sz indeed meta m in0 N infer meta n in0 N infer isSeries( meta fx , meta sz ) infer isSeries( meta fy , meta sz ) infer meta m = meta n infer meta fx = meta fy infer [ meta fx ; meta m ] = [ meta fy ; meta n ] end lemma end math ] "
" [ math system Q proof of lemma sameSeries(Gen) reads any term meta m comma meta n comma meta fx comma meta fy comma meta sz end line line ell big a premise meta m in0 N end line line ell big x premise meta n in0 N end line line ell big b premise isSeries( meta fx , meta sz ) end line line ell big c premise isSeries( meta fy , meta sz ) end line line ell a premise meta m = meta n end line line ell b premise meta fx = meta fy end line line ell e because lemma memberOfSeries modus ponens ell big a modus ponens ell big b indeed (o meta m , [ meta fx ; meta m ] ) in0 meta fx end line line ell f because lemma set equality nec condition(1) modus ponens ell b modus ponens ell e indeed (o meta m , [ meta fx ; meta m ] ) in0 meta fy end line line ell g because lemma memberOfSeries modus ponens ell big x modus ponens ell big c indeed (o meta n , [ meta fy ; meta n ] ) in0 meta fy end line because lemma uniqueMember modus ponens ell big c modus ponens ell f modus ponens ell g modus ponens ell a indeed [ meta fx ; meta m ] = [ meta fy ; meta n ] qed end math ] "
" [ math in theory system Q lemma lemma equalsSameF says for all terms meta fx comma meta fy indeed meta fx = meta fy infer meta fx sameF meta fy end lemma end math ] "
" [ math system Q proof of lemma equalsSameF reads block any term meta fx comma meta fy end line line ell x premise meta fx = meta fy end line line ell a premise 0 < object ep end line line ell b premise 0 <= object m end line line ell e because axiom natType indeed object m in0 N end line line ell g because axiom seriesType indeed isSeries( meta fx , Q ) end line line ell h because axiom seriesType indeed isSeries( meta fy , Q ) end line line ell i because lemma eqReflexivity indeed object m = object m end line line ell j because lemma sameSeries(Gen) modus ponens ell e modus ponens ell e modus ponens ell g modus ponens ell h modus ponens ell i modus ponens ell x indeed [ meta fx ; object m ] = [ meta fy ; object m ] end line line ell y because lemma positiveToLeft(Eq)(1 term) modus ponens ell j indeed [ meta fx ; object m ] - [ meta fy ; object m ] = 0 end line line ell z because lemma sameNumerical modus ponens ell y indeed | [ meta fx ; object m ] - [ meta fy ; object m ] | = | 0 | end line line ell u because lemma |0|=0 indeed | 0 | = 0 end line line ell c because lemma eqTransitivity modus ponens ell z modus ponens ell u indeed | [ meta fx ; object m ] - [ meta fy ; object m ] | = 0 end line line ell d because lemma eqSymmetry modus ponens ell c indeed 0 = | [ meta fx ; object m ] - [ meta fy ; object m ] | end line because lemma subLessLeft modus ponens ell d modus ponens ell a indeed | [ meta fx ; object m ] - [ meta fy ; object m ] | < object ep 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 0 < object ep imply 0 <= object m imply | [ meta fx ; object m ] - [ meta fy ; object m ] | < object ep end line line ell x premise meta fx = meta fy end line line ell y because 1rule mp modus ponens ell a modus ponens ell x indeed 0 < object ep imply 0 <= object m imply | [ meta fx ; object m ] - [ meta fy ; object m ] | < object ep end line line ell b because 1rule gen modus ponens ell y indeed for all object m indeed ( 0 < object ep imply 0 <= object m imply | [ meta fx ; object m ] - [ meta fy ; object m ] | < object ep ) end line line ell c because pred lemma intro exist at 0 modus ponens ell b 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 gen modus ponens ell c indeed 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 d indeed meta fx sameF meta fy 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 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 ] "
" [ 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 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 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 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 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 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 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 ] "
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 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 ] "
(*** 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 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 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 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 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 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 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 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 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 ( 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 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 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 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 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 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 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 (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 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 timesF(Sym) says for all terms meta m comma meta fx comma meta fy indeed [ meta fx ; meta m ] * [ meta fy ; meta m ] = [ meta fx *f meta fy ; meta m ] end lemma end math ] "
" [ math system Q proof of lemma timesF(Sym) reads any term meta m comma meta fx comma meta fy end line line ell a because lemma timesF indeed [ meta fx *f meta fy ; meta m ] = [ meta fx ; meta m ] * [ meta fy ; meta m ] end line because lemma eqSymmetry modus ponens ell a indeed [ meta fx ; meta m ] * [ meta fy ; meta m ] = [ meta fx *f 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 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 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 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 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 exist0 object op1 indeed exist0 object op2 indeed 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 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 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 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 ] "
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" [ 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 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 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 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 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 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 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 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 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
" [ 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
" [ 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
" [ 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
" [ 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
" [ 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
" [ 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 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 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 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 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 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 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 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 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 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 equalsSameF 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 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 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 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 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
" [ 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
" [ 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
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" [ 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 00 << R( meta fz ) or0 00 == R( meta fz ) end line line ell g because 1rule repetition modus ponens ell e indeed 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 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 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 for all object f1 comma object f2 comma object f3 comma object f4 indeed ( (o object f1 , object f2 ) in0 1f/ meta fx imply (o object f3 , object f4 ) in0 1f/ 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 1f/ meta fx end line line ell b premise (o object f3 , object f4 ) in0 1f/ 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 (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 (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 1f/ meta fx imply (o object f3 , object f4 ) in0 1f/ 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 1f/ 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 1f/ 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 1f/ 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 1f/ 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 1f/ 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 1f/ 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 1f/ 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 1f/ meta fx ) qed end math ] "
" [ math in theory system Q lemma lemma reciprocalIsRationalSeries says for all terms meta fx indeed isSeries( 1f/ 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( 1f/ 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 1f/ meta fx imply (o object f3 , object f4 ) in0 1f/ 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 1f/ meta fx ) end line because lemma toSeries modus ponens ell a modus ponens ell b modus ponens ell c indeed isSeries( 1f/ meta fx , Q ) qed end math ] "
" [ math in theory system Q lemma lemma reciprocalF nonzero says for all terms meta m comma meta m1 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 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( 1f/ meta fx , Q ) end line line ell c because lemma memberOfSeries modus ponens ell x modus ponens ell b indeed (o meta m , [ 1f/ meta fx ; meta m ] ) in0 1f/ meta fx end line line ell d because 1rule repetition modus ponens ell c indeed (o meta m , [ 1f/ 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 , [ 1f/ meta fx ; meta m ] ) = (o meta m1 , 1/ [ meta fx ; meta m1 ] ) ) or0 ( [ meta fx ; meta m1 ] = 0 and0 (o meta m , [ 1f/ 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 , [ 1f/ meta fx ; meta m ] ) = (o meta m1 , 1/ [ meta fx ; meta m1 ] ) ) or0 ( [ meta fx ; meta m1 ] = 0 and0 (o meta m , [ 1f/ 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 , [ 1f/ 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 , [ 1f/ 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 , [ 1f/ 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 [ 1f/ 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 [ 1f/ 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 , [ 1f/ meta fx ; meta m ] ) = (o meta m1 , 1/ [ meta fx ; meta m1 ] ) ) or0 ( [ meta fx ; meta m1 ] = 0 and0 (o meta m , [ 1f/ meta fx ; meta m ] ) = (o meta m1 , 0 ) ) imply [ 1f/ 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 , [ 1f/ meta fx ; meta m ] ) = (o meta m1 , 1/ [ meta fx ; meta m1 ] ) ) or0 ( [ meta fx ; meta m1 ] = 0 and0 (o meta m , [ 1f/ meta fx ; meta m ] ) = (o meta m1 , 0 ) ) imply [ 1f/ 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 [ 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 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 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
" [ math system Q proof of lemma switchTerms(x-y
" [ math in theory system Q lemma lemma switchTerms(x
" [ math system Q proof of lemma switchTerms(x
" [ 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 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
" [ math system Q proof of lemma fromNot
" [ math in theory system Q lemma lemma fromNot
" [ math system Q proof of lemma fromNot
" [ math in theory system Q lemma lemma fromNot
" [ math system Q proof of lemma fromNot
" [ math in theory system Q lemma lemma fromNot
" [ math system Q proof of lemma fromNot
" [ math in theory system Q lemma lemma fromNot
" [ math system Q proof of lemma fromNot
" [ 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 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 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 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 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 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
" [ math in theory system Q lemma lemma toLess(F) says 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
" [ 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 not0 parenthesis meta fx
" [ 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 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 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
" [ 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 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 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 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 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 ] "
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" [ math in theory system Q lemma lemma leqReflexivity(R) says for all terms meta fx indeed R( meta fx ) <<== R( meta fx ) end lemma end math ] "
" [ math system Q proof of lemma leqReflexivity(R) reads any term meta fx end line line ell a because lemma eqReflexivity indeed R( meta fx ) = R( meta fx ) end line because lemma eqLeq(R) modus ponens ell a indeed R( meta fx ) <<== R( meta fx ) 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 end math ] "
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\appendix
" [ flush left math priority table preassociative priority sup equal priority base equal priority bracket x end bracket equal priority big bracket x end bracket equal priority math x end math equal priority flush left x end left equal priority var x equal priority var y equal priority var z equal priority proclaim x as x end proclaim equal priority define x of x as x end define equal priority pyk equal priority tex equal priority tex name equal priority priority equal priority x equal priority true equal priority if x then x else x end if equal priority introduce x of x as x end introduce equal priority value equal priority claim equal priority bottom equal priority function f of x end function equal priority identity x end identity equal priority false equal priority untagged zero equal priority untagged one equal priority untagged two equal priority untagged three equal priority untagged four equal priority untagged five equal priority untagged six equal priority untagged seven equal priority untagged eight equal priority untagged nine equal priority zero equal priority one equal priority two equal priority three equal priority four equal priority five equal priority six equal priority seven equal priority eight equal priority nine equal priority var a equal priority var b equal priority var c equal priority var d equal priority var e equal priority var f equal priority var g equal priority var h equal priority var i equal priority var j equal priority var k equal priority var l equal priority var m equal priority var n equal priority var o equal priority var p equal priority var q equal priority var r equal priority var s equal priority var t equal priority var u equal priority var v equal priority var w equal priority tagged parenthesis x end tagged equal priority tagged if x then x else x end if equal priority array x is x end array equal priority left equal priority center equal priority right equal priority empty equal priority substitute x set x to x end substitute equal priority map tag x end tag equal priority raw map untag x end untag equal priority map untag x end untag equal priority normalizing untag x end untag equal priority apply x to x end apply equal priority apply one x to x end apply equal priority identifier x end identifier equal priority identifier one x plus id x end identifier equal priority array plus x and x end plus equal priority array remove x array x level x end remove equal priority array put x value x array x level x end put equal priority array add x value x index x value x level x end add equal priority bit x of x end bit equal priority bit one x of x end bit equal priority example rack equal priority vector hook equal priority bibliography hook equal priority dictionary hook equal priority body hook equal priority codex hook equal priority expansion hook equal priority code hook equal priority cache hook equal priority diagnose hook equal priority pyk aspect equal priority tex aspect equal priority texname aspect equal priority value aspect equal priority message aspect equal priority macro aspect equal priority definition aspect equal priority unpack aspect equal priority claim aspect equal priority priority aspect equal priority lambda identifier equal priority apply identifier equal priority true identifier equal priority if identifier equal priority quote identifier equal priority proclaim identifier equal priority define identifier equal priority introduce identifier equal priority hide identifier equal priority pre identifier equal priority post identifier equal priority eval x stack x cache x end eval equal priority eval two x ref x id x stack x cache x end eval equal priority eval three x function x stack x cache x end eval equal priority eval four x arguments x stack x cache x end eval equal priority lookup x stack x default x end lookup equal priority abstract x term x stack x cache x end abstract equal priority quote x end quote equal priority expand x state x cache x end expand equal priority expand two x definition x state x cache x end expand equal priority expand list x state x cache x end expand equal priority macro equal priority macro state equal priority zip x with x end zip equal priority assoc one x address x index x end assoc equal priority protect x end protect equal priority self equal priority macro define x as x end define equal priority value define x as x end define equal priority intro define x as x end define equal priority pyk define x as x end define equal priority tex define x as x end define equal priority tex name define x as x end define equal priority priority table x end table equal priority macro define one equal priority macro define two x end define equal priority macro define three x end define equal priority macro define four x state x cache x definition x end define equal priority state expand x state x cache x end expand equal priority quote expand x term x stack x end expand equal priority quote expand two x term x stack x end expand equal priority quote expand three x term x stack x value x end expand equal priority quote expand star x term x stack x end expand equal priority parenthesis x end parenthesis equal priority big parenthesis x end parenthesis equal priority display x end display equal priority statement x end statement equal priority spying test x end test equal priority false spying test x end test equal priority aspect x subcodex x end aspect equal priority aspect x term x cache x end aspect equal priority tuple x end tuple equal priority tuple one x end tuple equal priority tuple two x end tuple equal priority let two x apply x end let equal priority let one x apply x end let equal priority claim define x as x end define equal priority checker equal priority check x cache x end check equal priority check two x cache x def x end check equal priority check three x cache x def x end check equal priority check list x cache x end check equal priority check list two x cache x value x end check equal priority test x end test equal priority false test x end test equal priority raw test x end test equal priority message equal priority message define x as x end define equal priority the statement aspect equal priority statement equal priority statement define x as x end define equal priority example axiom equal priority example scheme equal priority example rule equal priority absurdity equal priority contraexample equal priority example theory primed equal priority example lemma equal priority metavar x end metavar equal priority meta a equal priority meta b equal priority meta c equal priority meta d equal priority meta e equal priority meta f equal priority meta g equal priority meta h equal priority meta i equal priority meta j equal priority meta k equal priority meta l equal priority meta m equal priority meta n equal priority meta o equal priority meta p equal priority meta q equal priority meta r equal priority meta s equal priority meta t equal priority meta u equal priority meta v equal priority meta w equal priority meta x equal priority meta y equal priority meta z equal priority sub x set x to x end sub equal priority sub star x set x to x end sub equal priority the empty set equal priority example remainder equal priority make visible x end visible equal priority intro x index x pyk x tex x end intro equal priority intro x pyk x tex x end intro equal priority error x term x end error equal priority error two x term x end error equal priority proof x term x cache x end proof equal priority proof two x term x end proof equal priority sequent eval x term x end eval equal priority seqeval init x term x end eval equal priority seqeval modus x term x end eval equal priority seqeval modus one x term x sequent x end eval equal priority seqeval verify x term x end eval equal priority seqeval verify one x term x sequent x end eval equal priority sequent eval plus x term x end eval equal priority seqeval plus one x term x sequent x end eval equal priority seqeval minus x term x end eval equal priority seqeval minus one x term x sequent x end eval equal priority seqeval deref x term x end eval equal priority seqeval deref one x term x sequent x end eval equal priority seqeval deref two x term x sequent x def x end eval equal priority seqeval at x term x end eval equal priority seqeval at one x term x sequent x end eval equal priority seqeval infer x term x end eval equal priority seqeval infer one x term x premise x sequent x end eval equal priority seqeval endorse x term x end eval equal priority seqeval endorse one x term x side x sequent x end eval equal priority seqeval est x term x end eval equal priority seqeval est one x term x name x sequent x end eval equal priority seqeval est two x term x name x sequent x def x end eval equal priority seqeval all x term x end eval equal priority seqeval all one x term x variable x sequent x end eval equal priority seqeval cut x term x end eval equal priority seqeval cut one x term x forerunner x end eval equal priority seqeval cut two x term x forerunner x sequent x end eval equal priority computably true x end true equal priority claims x cache x ref x end claims equal priority claims two x cache x ref x end claims equal priority the proof aspect equal priority proof equal priority lemma x says x end lemma equal priority proof of x reads x end proof equal priority in theory x lemma x says x end lemma equal priority in theory x antilemma x says x end antilemma equal priority in theory x rule x says x end rule equal priority in theory x antirule x says x end antirule equal priority verifier equal priority verify one x end verify equal priority verify two x proofs x end verify equal priority verify three x ref x sequents x diagnose x end verify equal priority verify four x premises x end verify equal priority verify five x ref x array x sequents x end verify equal priority verify six x ref x list x sequents x end verify equal priority verify seven x ref x id x sequents x end verify equal priority cut x and x end cut equal priority head x end head equal priority tail x end tail equal priority rule one x theory x end rule equal priority rule x subcodex x end rule equal priority rule tactic equal priority plus x and x end plus equal priority theory x end theory equal priority theory two x cache x end theory equal priority theory three x name x end theory equal priority theory four x name x sum x end theory equal priority example axiom lemma primed equal priority example scheme lemma primed equal priority example rule lemma primed equal priority contraexample lemma primed equal priority example axiom lemma equal priority example scheme lemma equal priority example rule lemma equal priority contraexample lemma equal priority example theory equal priority ragged right equal priority ragged right expansion equal priority parameter term x stack x seed x end parameter equal priority parameter term star x stack x seed x end parameter equal priority instantiate x with x end instantiate equal priority instantiate star x with x end instantiate equal priority occur x in x substitution x end occur equal priority occur star x in x substitution x end occur equal priority unify x with x substitution x end unify equal priority unify star x with x substitution x end unify equal priority unify two x with x substitution x end unify equal priority ell a equal priority ell b equal priority ell c equal priority ell d equal priority ell e equal priority ell f equal priority ell g equal priority ell h equal priority ell i equal priority ell j equal priority ell k equal priority ell l equal priority ell m equal priority ell n equal priority ell o equal priority ell p equal priority ell q equal priority ell r equal priority ell s equal priority ell t equal priority ell u equal priority ell v equal priority ell w equal priority ell x equal priority ell y equal priority ell z equal priority ell big a equal priority ell big b equal priority ell big c equal priority ell big d equal priority ell big e equal priority ell big f equal priority ell big g equal priority ell big h equal priority ell big i equal priority ell big j equal priority ell big k equal priority ell big l equal priority ell big m equal priority ell big n equal priority ell big o equal priority ell big p equal priority ell big q equal priority ell big r equal priority ell big s equal priority ell big t equal priority ell big u equal priority ell big v equal priority ell big w equal priority ell big x equal priority ell big y equal priority ell big z equal priority ell dummy equal priority sequent reflexivity equal priority tactic reflexivity equal priority sequent commutativity equal priority tactic commutativity equal priority the tactic aspect equal priority tactic equal priority tactic define x as x end define equal priority proof expand x state x cache x end expand equal priority proof expand list x state x cache x end expand equal priority proof state equal priority conclude one x cache x end conclude equal priority conclude two x proves x cache x end conclude equal priority conclude three x proves x lemma x substitution x end conclude equal priority conclude four x lemma x end conclude equal priority check equal priority general macro define x as x end define equal priority make root visible x end visible equal priority sequent example axiom equal priority sequent example rule equal priority sequent example contradiction equal priority sequent example theory equal priority sequent example lemma equal priority set x end set equal priority object var x end var equal priority object a equal priority object b equal priority object c equal priority object d equal priority object e equal priority object f equal priority object g equal priority object h equal priority object i equal priority object j equal priority object k equal priority object l equal priority object m equal priority object n equal priority object o equal priority object p equal priority object q equal priority object r equal priority object s equal priority object t equal priority object u equal priority object v equal priority object w equal priority object x equal priority object y equal priority object z equal priority sub x is x where x is x end sub equal priority sub zero x is x where x is x end sub equal priority sub one x is x where x is x end sub equal priority sub star x is x where x is x end sub equal priority deduction x conclude x end deduction equal priority deduction zero x conclude x end deduction equal priority deduction one x conclude x condition x end deduction equal priority deduction two x conclude x condition x end deduction equal priority deduction three x conclude x condition x bound x end deduction equal priority deduction four x conclude x condition x bound x end deduction equal priority deduction four star x conclude x condition x bound x end deduction equal priority deduction five x condition x bound x end deduction equal priority deduction six x conclude x exception x bound x end deduction equal priority deduction six star x conclude x exception x bound x end deduction equal priority deduction seven x end deduction equal priority deduction eight x bound x end deduction equal priority deduction eight star x bound x end deduction equal priority system s equal priority double negation equal priority rule mp equal priority rule gen equal priority deduction equal priority axiom s one equal priority axiom s two equal priority axiom s three equal priority axiom s four equal priority axiom s five equal priority axiom s six equal priority axiom s seven equal priority axiom s eight equal priority axiom s nine equal priority repetition equal priority lemma a one equal priority lemma a two equal priority lemma a four equal priority lemma a five equal priority prop three two a equal priority prop three two b equal priority prop three two c equal priority prop three two d equal priority prop three two e one equal priority prop three two e two equal priority prop three two e equal priority prop three two f one equal priority prop three two f two equal priority prop three two f equal priority prop three two g one equal priority prop three two g two equal priority prop three two g equal priority prop three two h one equal priority prop three two h two equal priority prop three two h equal priority block one x state x cache x end block equal priority block two x end block equal priority kvanti equal priority lemma uniqueMember equal priority lemma uniqueMember(Type) equal priority lemma sameSeries equal priority lemma a4 equal priority lemma sameMember equal priority 1rule Qclosed(Addition) equal priority 1rule Qclosed(Multiplication) equal priority 1rule fromCartProd(1) equal priority 1rule fromCartProd(2) equal priority constantRationalSeries( x ) equal priority cartProd( x , x ) equal priority P( x ) equal priority binaryUnion( x , x ) equal priority setOfRationalSeries equal priority isSubset( x , x ) equal priority (p x , x ) equal priority (s x ) equal priority cdots equal priority object-var equal priority ex-var equal priority ph-var equal priority vaerdi equal priority variabel equal priority op x end op equal priority op2 x comma x end op2 equal priority define-equal x comma x end equal equal priority contains-empty x end empty equal priority Nat( x ) equal priority 1deduction x conclude x end 1deduction equal priority 1deduction zero x conclude x end 1deduction equal priority 1deduction side x conclude x condition x end 1deduction equal priority 1deduction one x conclude x condition x end 1deduction equal priority 1deduction two x conclude x condition x end 1deduction equal priority 1deduction three x conclude x condition x bound x end 1deduction equal priority 1deduction four x conclude x condition x bound x end 1deduction equal priority 1deduction four star x conclude x condition x bound x end 1deduction equal priority 1deduction five x condition x bound x end 1deduction equal priority 1deduction six x conclude x exception x bound x end 1deduction equal priority 1deduction six star x conclude x exception x bound x end 1deduction equal priority 1deduction seven x end 1deduction equal priority 1deduction eight x bound x end 1deduction equal priority 1deduction eight star x bound x end 1deduction equal priority ex1 equal priority ex2 equal priority ex3 equal priority ex10 equal priority ex20 equal priority existential var x end var equal priority x is existential var equal priority exist-sub x is x where x is x end sub equal priority exist-sub0 x is x where x is x end sub equal priority exist-sub1 x is x where x is x end sub equal priority exist-sub* x is x where x is x end sub equal priority ph1 equal priority ph2 equal priority ph3 equal priority placeholder-var x end var equal priority x is placeholder-var equal priority ph-sub x is x where x is x end sub equal priority ph-sub0 x is x where x is x end sub equal priority ph-sub1 x is x where x is x end sub equal priority ph-sub* x is x where x is x end sub equal priority meta-sub x is x where x is x end sub equal priority meta-sub1 x is x where x is x end sub equal priority meta-sub* x is x where x is x end sub equal priority var big set equal priority object big set equal priority meta big set equal priority zermelo empty set equal priority system Q equal priority 1rule mp equal priority 1rule gen equal priority 1rule repetition equal priority 1rule ad absurdum equal priority 1rule deduction equal priority 1rule exist intro equal priority axiom extensionality equal priority axiom empty set equal priority axiom pair definition equal priority axiom union definition equal priority axiom power definition equal priority axiom separation definition equal priority prop lemma add double neg equal priority prop lemma remove double neg equal priority prop lemma and commutativity equal priority prop lemma auto imply equal priority prop lemma contrapositive equal priority prop lemma first conjunct equal priority prop lemma second conjunct equal priority prop lemma from contradiction equal priority prop lemma from disjuncts equal priority prop lemma iff commutativity equal priority prop lemma iff first equal priority prop lemma iff second equal priority prop lemma imply transitivity equal priority prop lemma join conjuncts equal priority prop lemma mp2 equal priority prop lemma mp3 equal priority prop lemma mp4 equal priority prop lemma mp5 equal priority prop lemma mt equal priority prop lemma negative mt equal priority prop lemma technicality equal priority prop lemma weakening equal priority prop lemma weaken or first equal priority prop lemma weaken or second equal priority lemma formula2pair equal priority lemma pair2formula equal priority lemma formula2union equal priority lemma union2formula equal priority lemma formula2separation equal priority lemma separation2formula equal priority lemma formula2power equal priority lemma subset in power set equal priority lemma power set is subset0 equal priority lemma power set is subset equal priority lemma power set is subset0-switch equal priority lemma power set is subset-switch equal priority lemma set equality suff condition equal priority lemma set equality suff condition(t)0 equal priority lemma set equality suff condition(t) equal priority lemma set equality skip quantifier equal priority lemma set equality nec condition equal priority lemma reflexivity0 equal priority lemma reflexivity equal priority lemma symmetry0 equal priority lemma symmetry equal priority lemma transitivity0 equal priority lemma transitivity equal priority lemma er is reflexive equal priority lemma er is symmetric equal priority lemma er is transitive equal priority lemma empty set is subset equal priority lemma member not empty0 equal priority lemma member not empty equal priority lemma unique empty set0 equal priority lemma unique empty set equal priority lemma ==Reflexivity equal priority lemma ==Symmetry equal priority lemma ==Transitivity0 equal priority lemma ==Transitivity equal priority lemma transfer ~is0 equal priority lemma transfer ~is equal priority lemma pair subset0 equal priority lemma pair subset1 equal priority lemma pair subset equal priority lemma same pair equal priority lemma same singleton equal priority lemma union subset equal priority lemma same union equal priority lemma separation subset equal priority lemma same separation equal priority lemma same binary union equal priority lemma intersection subset equal priority lemma same intersection equal priority lemma auto member equal priority lemma eq-system not empty0 equal priority lemma eq-system not empty equal priority lemma eq subset0 equal priority lemma eq subset equal priority lemma equivalence nec condition0 equal priority lemma equivalence nec condition equal priority lemma none-equivalence nec condition0 equal priority lemma none-equivalence nec condition1 equal priority lemma none-equivalence nec condition equal priority lemma equivalence class is subset equal priority lemma equivalence classes are disjoint equal priority lemma all disjoint equal priority lemma all disjoint-imply equal priority lemma bs subset union(bs/r) equal priority lemma union(bs/r) subset bs equal priority lemma union(bs/r) is bs equal priority theorem eq-system is partition equal priority var x1 equal priority var x2 equal priority var y1 equal priority var y2 equal priority var v1 equal priority var v2 equal priority var v3 equal priority var v4 equal priority var v2n equal priority var m1 equal priority var m2 equal priority var n1 equal priority var n2 equal priority var n3 equal priority var ep equal priority var ep1 equal priority var ep2 equal priority var fep equal priority var fx equal priority var fy equal priority var fz equal priority var fu equal priority var fv equal priority var fw equal priority var rx equal priority var ry equal priority var rz equal priority var ru equal priority var sx equal priority var sx1 equal priority var sy equal priority var sy1 equal priority var sz equal priority var sz1 equal priority var su equal priority var su1 equal priority var fxs equal priority var fys equal priority var crs1 equal priority var f1 equal priority var f2 equal priority var f3 equal priority var f4 equal priority var op1 equal priority var op2 equal priority var r1 equal priority var s1 equal priority var s2 equal priority meta x1 equal priority meta x2 equal priority meta y1 equal priority meta y2 equal priority meta v1 equal priority meta v2 equal priority meta v3 equal priority meta v4 equal priority meta v2n equal priority meta m1 equal priority meta m2 equal priority meta n1 equal priority meta n2 equal priority meta n3 equal priority meta ep equal priority meta ep1 equal priority meta ep2 equal priority meta fx equal priority meta fy equal priority meta fz equal priority meta fu equal priority meta fv equal priority meta fw equal priority meta fep equal priority meta rx equal priority meta ry equal priority meta rz equal priority meta ru equal priority meta sx equal priority meta sx1 equal priority meta sy equal priority meta sy1 equal priority meta sz equal priority meta sz1 equal priority meta su equal priority meta su1 equal priority meta fxs equal priority meta fys equal priority meta f1 equal priority meta f2 equal priority meta f3 equal priority meta f4 equal priority meta op1 equal priority meta op2 equal priority meta r1 equal priority meta s1 equal priority meta s2 equal priority object ep equal priority object crs1 equal priority object f1 equal priority object f2 equal priority object f3 equal priority object f4 equal priority object n1 equal priority object n2 equal priority object op1 equal priority object op2 equal priority object r1 equal priority object s1 equal priority object s2 equal priority ph4 equal priority ph5 equal priority ph6 equal priority NAT equal priority RATIONAL_SERIES equal priority SERIES equal priority setOfReals equal priority setOfFxs equal priority N equal priority Q equal priority X equal priority xs equal priority xsF equal priority ysF equal priority us equal priority usF equal priority 0 equal priority 1 equal priority (-1) equal priority 2 equal priority 3 equal priority 1/2 equal priority 1/3 equal priority 2/3 equal priority 0f equal priority 1f equal priority 00 equal priority 01 equal priority (--01) equal priority 02 equal priority 01//02 equal priority lemma plusAssociativity(R) equal priority lemma plusAssociativity(R)XX equal priority lemma plus0(R) equal priority lemma negative(R) equal priority lemma times1(R) equal priority lemma lessAddition(R) equal priority lemma plusCommutativity(R) equal priority lemma leqAntisymmetry(R) equal priority lemma leqTransitivity(R) equal priority lemma leqAddition(R) equal priority lemma distribution(R) equal priority axiom a4 equal priority axiom induction equal priority axiom equality equal priority axiom eqLeq equal priority axiom eqAddition equal priority axiom eqMultiplication equal priority axiom QisClosed(reciprocal) equal priority lemma QisClosed(reciprocal) equal priority axiom QisClosed(negative) equal priority lemma QisClosed(negative) equal priority axiom leqReflexivity equal priority axiom leqAntisymmetry equal priority axiom leqTransitivity equal priority axiom leqTotality equal priority axiom leqAddition equal priority axiom leqMultiplication equal priority axiom plusAssociativity equal priority axiom plusCommutativity equal priority axiom negative equal priority axiom plus0 equal priority axiom timesAssociativity equal priority axiom timesCommutativity equal priority axiom reciprocal equal priority axiom times1 equal priority axiom distribution equal priority axiom 0not1 equal priority lemma eqLeq(R) equal priority lemma timesAssociativity(R) equal priority lemma timesCommutativity(R) equal priority 1rule adhoc sameR equal priority lemma separation2formula(1) equal priority lemma separation2formula(2) equal priority axiom cauchy equal priority axiom plusF equal priority axiom reciprocalF equal priority 1rule from== equal priority 1rule to== equal priority 1rule fromInR equal priority lemma plusR(Sym) equal priority axiom reciprocalR equal priority 1rule lessMinus1(N) equal priority axiom nonnegative(N) equal priority axiom US0 equal priority 1rule nextXS(upperBound) equal priority 1rule nextXS(noUpperBound) equal priority 1rule nextUS(upperBound) equal priority 1rule nextUS(noUpperBound) equal priority 1rule expZero equal priority 1rule expPositive equal priority 1rule expZero(R) equal priority 1rule expPositive(R) equal priority 1rule base(1/2)Sum zero equal priority 1rule base(1/2)Sum positive equal priority 1rule UStelescope zero equal priority 1rule UStelescope positive equal priority 1rule adhoc eqAddition(R) equal priority 1rule fromLimit equal priority 1rule toUpperBound equal priority 1rule fromUpperBound equal priority axiom USisUpperBound equal priority axiom 0not1(R) equal priority 1rule expUnbounded equal priority 1rule fromLeq(Advanced)(N) equal priority 1rule fromLeastUpperBound equal priority 1rule toLeastUpperBound equal priority axiom XSisNotUpperBound equal priority axiom ysFGreater equal priority axiom ysFLess equal priority 1rule smallInverse equal priority axiom natType equal priority axiom rationalType equal priority axiom seriesType equal priority axiom max equal priority axiom numerical equal priority axiom numericalF equal priority axiom memberOfSeries equal priority prop lemma doubly conditioned join conjuncts equal priority prop lemma imply negation equal priority prop lemma tertium non datur equal priority prop lemma from negated imply equal priority prop lemma to negated imply equal priority prop lemma from negated double imply equal priority prop lemma from negated and equal priority prop lemma from negated or equal priority prop lemma to negated or equal priority prop lemma from negations equal priority prop lemma from three disjuncts equal priority prop lemma from two times two disjuncts equal priority prop lemma negate first disjunct equal priority prop lemma negate second disjunct equal priority prop lemma expand disjuncts equal priority lemma set equality nec condition(1) equal priority lemma set equality nec condition(2) equal priority lemma lessLeq(R) equal priority lemma memberOfSeries equal priority lemma memberOfSeries(Type) equal priority prop lemma to negated and(1) equal priority lemma uniqueNegative equal priority lemma doubleMinus equal priority lemma minusNegated equal priority lemma eqReflexivity equal priority lemma eqSymmetry equal priority lemma eqTransitivity equal priority lemma eqTransitivity4 equal priority lemma eqTransitivity5 equal priority lemma eqTransitivity6 equal priority lemma addEquations equal priority lemma subtractEquations equal priority lemma subtractEquationsLeft equal priority lemma multiplyEquations equal priority lemma eqNegated equal priority lemma positiveToRight(Eq) equal priority lemma positiveToLeft(Eq)(1 term) equal priority lemma negativeToLeft(Eq) equal priority lemma nonreciprocalToRight(Eq)(1 term) equal priority lemma plusAssociativity(4 terms) equal priority lemma lessNeq equal priority lemma neqSymmetry equal priority lemma neqNegated equal priority lemma subNeqRight equal priority lemma subNeqLeft equal priority lemma negativeToRight(Neq)(1 term) equal priority lemma neqAddition equal priority lemma neqMultiplication equal priority lemma nonzeroProduct(2) equal priority lemma UStelescope(+1) equal priority lemma telescopeBound base equal priority lemma telescopeBound indu equal priority lemma telescopeBound equal priority lemma intervalSize base equal priority lemma intervalSize indu equal priority lemma intervalSize equal priority lemma XSlessUS equal priority lemma USdecreasing(+1) equal priority lemma closeUS equal priority lemma closeUS(n+1) equal priority pred lemma allNegated(Imply) equal priority pred lemma existNegated(Imply) equal priority pred lemma intro exist helper equal priority pred lemma intro exist equal priority pred lemma exist mp equal priority pred lemma exist mp2 equal priority pred lemma 2exist mp equal priority pred lemma 2exist mp2 equal priority pred lemma EAE mp equal priority pred lemma addAll equal priority pred lemma addExist helper1 equal priority pred lemma addExist helper2 equal priority pred lemma addExist equal priority pred lemma addExist(SimpleAnt) equal priority pred lemma addExist(Simple) equal priority pred lemma addEAE equal priority pred lemma AEAnegated equal priority pred lemma EEAnegated equal priority lemma induction equal priority lemma leqAntisymmetry equal priority lemma leqTransitivity equal priority lemma leqAddition equal priority lemma leqMultiplication equal priority lemma reciprocal equal priority lemma equality equal priority lemma eqLeq equal priority lemma eqAddition equal priority lemma eqMultiplication equal priority lemma leqMultiplicationLeft equal priority lemma leqLessEq equal priority lemma lessLeq equal priority lemma from leqGeq equal priority lemma subLeqRight equal priority lemma subLeqLeft equal priority lemma leqPlus1 equal priority lemma positiveToRight(Leq) equal priority lemma positiveToRight(Leq)(1 term) equal priority lemma negativeToRight(Leq) equal priority lemma positiveToLeft(Leq) equal priority lemma negativeToLeft(Leq) equal priority lemma negativeToLeft(Leq)(1 term) equal priority lemma leqAdditionLeft equal priority lemma leqSubtraction equal priority lemma leqSubtractionLeft equal priority lemma thirdGeq equal priority lemma leqNegated equal priority lemma addEquations(Leq) equal priority lemma multiplyEquations(Leq) equal priority lemma thirdGeqSeries equal priority lemma leqNeqLess equal priority lemma fromLess equal priority lemma toLess equal priority lemma fromNotLess equal priority lemma toNotLess equal priority lemma negativeLessPositive equal priority lemma leqLessTransitivity equal priority lemma lessLeqTransitivity equal priority lemma lessTransitivity equal priority lemma lessTotality equal priority lemma subLessRight equal priority lemma subLessLeft equal priority lemma switchTerms(x
\section{Pyk definitioner} \label{sec:pyk}
\begin{flushleft}
" [ math protect define pyk of prop lemma to negated and(1) as text "prop lemma to negated and(1)" end text end define linebreak define pyk of lemma uniqueNegative as text "lemma uniqueNegative" end text end define linebreak define pyk of lemma doubleMinus as text "lemma doubleMinus" end text end define linebreak define pyk of lemma minusNegated as text "lemma minusNegated" end text end define linebreak define pyk of lemma eqReflexivity as text "lemma eqReflexivity" end text end define linebreak define pyk of lemma eqSymmetry as text "lemma eqSymmetry" end text end define linebreak define pyk of lemma eqTransitivity as text "lemma eqTransitivity" end text end define linebreak define pyk of lemma eqTransitivity4 as text "lemma eqTransitivity4" end text end define linebreak define pyk of lemma eqTransitivity5 as text "lemma eqTransitivity5" end text end define linebreak define pyk of lemma eqTransitivity6 as text "lemma eqTransitivity6" end text end define linebreak define pyk of lemma addEquations as text "lemma addEquations" end text end define linebreak define pyk of lemma subtractEquations as text "lemma subtractEquations" end text end define linebreak define pyk of lemma subtractEquationsLeft as text "lemma subtractEquationsLeft" end text end define linebreak define pyk of lemma multiplyEquations as text "lemma multiplyEquations" end text end define linebreak define pyk of lemma eqNegated as text "lemma eqNegated" end text end define linebreak define pyk of lemma positiveToRight(Eq) as text "lemma positiveToRight(Eq)" end text end define linebreak define pyk of lemma positiveToLeft(Eq)(1 term) as text "lemma positiveToLeft(Eq)(1 term)" end text end define linebreak define pyk of lemma negativeToLeft(Eq) as text "lemma negativeToLeft(Eq)" end text end define linebreak define pyk of lemma nonreciprocalToRight(Eq)(1 term) as text "lemma nonreciprocalToRight(Eq)(1 term)" end text end define linebreak define pyk of lemma plusAssociativity(4 terms) as text "lemma plusAssociativity(4 terms)" end text end define linebreak define pyk of lemma lessNeq as text "lemma lessNeq" end text end define linebreak define pyk of lemma neqSymmetry as text "lemma neqSymmetry" end text end define linebreak define pyk of lemma neqNegated as text "lemma neqNegated" end text end define linebreak define pyk of lemma subNeqRight as text "lemma subNeqRight" end text end define linebreak define pyk of lemma subNeqLeft as text "lemma subNeqLeft" end text end define linebreak define pyk of lemma negativeToRight(Neq)(1 term) as text "lemma negativeToRight(Neq)(1 term)" end text end define linebreak define pyk of lemma neqAddition as text "lemma neqAddition" end text end define linebreak define pyk of lemma neqMultiplication as text "lemma neqMultiplication" end text end define linebreak define pyk of lemma nonzeroProduct(2) as text "lemma nonzeroProduct(2)" end text end define linebreak define pyk of lemma UStelescope(+1) as text "lemma UStelescope(+1)" end text end define linebreak define pyk of lemma telescopeBound base as text "lemma telescopeBound base" end text end define linebreak define pyk of lemma telescopeBound indu as text "lemma telescopeBound indu" end text end define linebreak define pyk of lemma telescopeBound as text "lemma telescopeBound" end text end define linebreak define pyk of lemma intervalSize base as text "lemma intervalSize base" end text end define linebreak define pyk of lemma intervalSize indu as text "lemma intervalSize indu" end text end define linebreak define pyk of lemma intervalSize as text "lemma intervalSize" end text end define linebreak define pyk of lemma XSlessUS as text "lemma XSlessUS" end text end define linebreak define pyk of lemma USdecreasing(+1) as text "lemma USdecreasing(+1)" end text end define linebreak define pyk of lemma closeUS as text "lemma closeUS" end text end define linebreak define pyk of lemma closeUS(n+1) as text "lemma closeUS(n+1)" end text end define linebreak define pyk of pred lemma allNegated(Imply) as text "pred lemma allNegated(Imply)" end text end define linebreak define pyk of pred lemma existNegated(Imply) as text "pred lemma existNegated(Imply)" end text end define linebreak define pyk of pred lemma intro exist helper as text "pred lemma intro exist helper" end text end define linebreak define pyk of pred lemma intro exist as text "pred lemma intro exist" end text end define linebreak define pyk of pred lemma exist mp as text "pred lemma exist mp" end text end define linebreak define pyk of pred lemma exist mp2 as text "pred lemma exist mp2" end text end define linebreak define pyk of pred lemma 2exist mp as text "pred lemma 2exist mp" end text end define linebreak define pyk of pred lemma 2exist mp2 as text "pred lemma 2exist mp2" end text end define linebreak define pyk of pred lemma EAE mp as text "pred lemma EAE mp" end text end define linebreak define pyk of pred lemma addAll as text "pred lemma addAll" end text end define linebreak define pyk of pred lemma addExist helper1 as text "pred lemma addExist helper1" end text end define linebreak define pyk of pred lemma addExist helper2 as text "pred lemma addExist helper2" end text end define linebreak define pyk of pred lemma addExist as text "pred lemma addExist" end text end define linebreak define pyk of pred lemma addExist(SimpleAnt) as text "pred lemma addExist(SimpleAnt)" end text end define linebreak define pyk of pred lemma addExist(Simple) as text "pred lemma addExist(Simple)" end text end define linebreak define pyk of pred lemma addEAE as text "pred lemma addEAE" end text end define linebreak define pyk of pred lemma AEAnegated as text "pred lemma AEAnegated" end text end define linebreak define pyk of pred lemma EEAnegated as text "pred lemma EEAnegated" end text end define linebreak define pyk of lemma induction as text "lemma induction" end text end define linebreak define pyk of lemma leqAntisymmetry as text "lemma leqAntisymmetry" end text end define linebreak define pyk of lemma leqTransitivity as text "lemma leqTransitivity" end text end define linebreak define pyk of lemma leqAddition as text "lemma leqAddition" end text end define linebreak define pyk of lemma leqMultiplication as text "lemma leqMultiplication" end text end define linebreak define pyk of lemma reciprocal as text "lemma reciprocal" end text end define linebreak define pyk of lemma equality as text "lemma equality" end text end define linebreak define pyk of lemma eqLeq as text "lemma eqLeq" end text end define linebreak define pyk of lemma eqAddition as text "lemma eqAddition" end text end define linebreak define pyk of lemma eqMultiplication as text "lemma eqMultiplication" end text end define linebreak define pyk of lemma leqMultiplicationLeft as text "lemma leqMultiplicationLeft" end text end define linebreak define pyk of lemma leqLessEq as text "lemma leqLessEq" end text end define linebreak define pyk of lemma lessLeq as text "lemma lessLeq" end text end define linebreak define pyk of lemma from leqGeq as text "lemma from leqGeq" end text end define linebreak define pyk of lemma subLeqRight as text "lemma subLeqRight" end text end define linebreak define pyk of lemma subLeqLeft as text "lemma subLeqLeft" end text end define linebreak define pyk of lemma leqPlus1 as text "lemma leqPlus1" end text end define linebreak define pyk of lemma positiveToRight(Leq) as text "lemma positiveToRight(Leq)" end text end define linebreak define pyk of lemma positiveToRight(Leq)(1 term) as text "lemma positiveToRight(Leq)(1 term)" end text end define linebreak define pyk of lemma negativeToRight(Leq) as text "lemma negativeToRight(Leq)" end text end define linebreak define pyk of lemma positiveToLeft(Leq) as text "lemma positiveToLeft(Leq)" end text end define linebreak define pyk of lemma negativeToLeft(Leq) as text "lemma negativeToLeft(Leq)" end text end define linebreak define pyk of lemma negativeToLeft(Leq)(1 term) as text "lemma negativeToLeft(Leq)(1 term)" end text end define linebreak define pyk of lemma leqAdditionLeft as text "lemma leqAdditionLeft" end text end define linebreak define pyk of lemma leqSubtraction as text "lemma leqSubtraction" end text end define linebreak define pyk of lemma leqSubtractionLeft as text "lemma leqSubtractionLeft" end text end define linebreak define pyk of lemma thirdGeq as text "lemma thirdGeq" end text end define linebreak define pyk of lemma leqNegated as text "lemma leqNegated" end text end define linebreak define pyk of lemma addEquations(Leq) as text "lemma addEquations(Leq)" end text end define linebreak define pyk of lemma multiplyEquations(Leq) as text "lemma multiplyEquations(Leq)" end text end define linebreak define pyk of lemma thirdGeqSeries as text "lemma thirdGeqSeries" end text end define linebreak define pyk of lemma leqNeqLess as text "lemma leqNeqLess" end text end define linebreak define pyk of lemma fromLess as text "lemma fromLess" end text end define linebreak define pyk of lemma toLess as text "lemma toLess" end text end define linebreak define pyk of lemma fromNotLess as text "lemma fromNotLess" end text end define linebreak define pyk of lemma toNotLess as text "lemma toNotLess" end text end define linebreak define pyk of lemma negativeLessPositive as text "lemma negativeLessPositive" end text end define linebreak define pyk of lemma leqLessTransitivity as text "lemma leqLessTransitivity" end text end define linebreak define pyk of lemma lessLeqTransitivity as text "lemma lessLeqTransitivity" end text end define linebreak define pyk of lemma lessTransitivity as text "lemma lessTransitivity" end text end define linebreak define pyk of lemma lessTotality as text "lemma lessTotality" end text end define linebreak define pyk of lemma subLessRight as text "lemma subLessRight" end text end define linebreak define pyk of lemma subLessLeft as text "lemma subLessLeft" end text end define linebreak define pyk of lemma switchTerms(x
\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.
(expARGH!) #2.
)" end define end math ] "
\item " [ math tex define lemma nonreciprocalToRight(Eq)(1 term) as "NonreciprocalToRight(Eq)(1 term)" 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) as "NonzeroProduct(2)" 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 UStelescope(+1) as "UStelescope(+1)" 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 intervalSize base 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 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 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 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)(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 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 as "UniqueNegative" end define end math ] "
\item " [ math tex define lemma doubleMinus as "DoubleMinus" end define end math ] "
\item " [ math tex define lemma leqMultiplicationLeft 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 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
\item " [ math tex define lemma switchTerms(x-y
\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 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 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) 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 as "ToNegatedAnd" end define end math ] "
\item " [ math tex define lemma eqTransitivity4 as "eqTransitivity4" end define end math ] "
\item " [ math tex define lemma sameFsymmetry as "SFsymmetry" end define end math ] "
\item " [ math tex define lemma sameFtransitivity as "SFtransitivity" 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 <
\item " [ math tex define lemma <
\item " [ math tex define lemma <<==Reflexivity 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
\item " [ math tex define lemma fromNot
\item " [ math tex define lemma fromNot
\item " [ math tex define lemma fromNot
\item " [ math tex define lemma fromNot
\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 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 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 ] "
\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 eqAdditionLeft as "EqAdditionLeft" end define end math ] "
\item " [ math tex define lemma eqMultiplicationLeft as "EqMultiplicationLeft" 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 sameSeries(Gen) as "SameSeries(Gen)" end define end math ] "
\item " [ math tex define lemma equalsSameF as "EqualsSameF" end define end math ] "
\item " [ math tex define lemma leqReflexivity(R) as "LeqReflexivity(R)" end define end math ] "
\end{list}
\end{document}
End of file
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